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  • Types of functionsWikipedia

  • Contents

    1 Algebraic function 11.1 Algebraic functions in one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.1.1 Introduction and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 The role of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    1.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    2 Almost periodic function 62.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    2.1.1 Uniform or Bohr or Bochner almost periodic functions . . . . . . . . . . . . . . . . . . . . 62.1.2 Stepanov almost periodic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.3 Weyl almost periodic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.4 Besicovitch almost periodic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.5 Almost periodic functions on a locally compact abelian group . . . . . . . . . . . . . . . . 8

    2.2 Quasiperiodic signals in audio and music synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    3 Antiholomorphic function 12

    4 Automorphic function 134.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    5 Baire function 145.1 Classication of Baire functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 Baire class 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3 Baire class 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.4 Baire class 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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    5.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    6 Baire one star function 166.1 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    7 Barrier function 177.1 Logarithmic barrier function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    7.1.1 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.1.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    7.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    8 Basis function 188.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

    8.1.1 Polynomial bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.1.2 Fourier basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

    8.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

    9 Bijection 199.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    9.2.1 Batting line-up of a baseball team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.2.2 Seats and students of a classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    9.3 More mathematical examples and some non-examples . . . . . . . . . . . . . . . . . . . . . . . . 219.4 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.5 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.6 Bijections and cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.8 Bijections and category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.9 Generalization to partial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.10 Contrast with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    10 Binary function 2510.1 Alternative denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.2 Example division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.3 Restrictions to ordinary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.4 Generalisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

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    10.5 Generalisations to ternary and other functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.6 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    11 Bochner measurable function 2711.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    12 Bounded function 2912.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

    13 Bounded type (mathematics) 3113.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3113.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3213.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3313.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

    14 Cauchy-continuous function 3414.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.3 Examples and non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.4 Generalisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

    15 Closed convex function 3615.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3615.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

    16 Coarse function 3716.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3716.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

    17 Coercive function 3817.1 Coercive vector elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3817.2 Coercive operators and forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3817.3 Norm-coercive mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3917.4 (Extended valued) coercive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3917.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

    18 Complex-valued function 4018.1 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4118.2 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4218.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4218.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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    19 Concave function 4319.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4319.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4419.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4419.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4419.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4519.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    20 Constant function 4620.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4620.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4620.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4820.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

    21 Constructible function 4921.1 Time-constructible denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4921.2 Space-constructible denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4921.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4921.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

    22 Continuous function 5122.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5122.2 Real-valued continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

    22.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5122.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5422.2.3 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5722.2.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5822.2.5 Directional and semi-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

    22.3 Continuous functions between metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6022.3.1 Uniform, Hlder and Lipschitz continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 61

    22.4 Continuous functions between topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 6122.4.1 Alternative denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6322.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6422.4.3 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6522.4.4 Dening topologies via continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . 65

    22.5 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6522.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6622.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6622.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

    23 Continuous functions on a compact Hausdor space 6823.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6823.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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    23.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    24 Convex function 7024.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7124.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7124.3 Convex function calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7224.4 Strongly convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

    24.4.1 Uniformly convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7524.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7624.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7624.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

    25 Elementary function 7725.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7725.2 Dierential algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7825.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7825.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7825.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

    26 Empty function 7926.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

    27 Even and odd functions 8027.1 Denition and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

    27.1.1 Even functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8127.1.2 Odd functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

    27.2 Some facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8327.2.1 Continuity and dierentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8327.2.2 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8327.2.3 Calculus properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

    27.3 Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8627.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8627.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8627.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

    28 Fabius function 8828.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

    29 Functional (mathematics) 9029.1 Functional details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

    29.1.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9029.1.2 Denite integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

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    29.1.3 Vector scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9229.1.4 Locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

    29.2 Functional equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9229.3 Functional derivative and functional integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9229.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9329.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

    30 Global analytic function 9430.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

    30.1.1 Sheaf-theoretic denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9430.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

    31 HardyLittlewood maximal function 9531.1 HardyLittlewood maximal inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9531.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9631.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9631.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9731.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

    32 Hermitian function 9932.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9932.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

    33 Homogeneous function 10133.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

    33.1.1 Linear functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10133.1.2 Homogeneous polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10133.1.3 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10233.1.4 Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

    33.2 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10333.2.1 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10333.2.2 Ane functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

    33.3 Positive homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10333.4 Homogeneous distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10433.5 Application to dierential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10433.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10433.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10533.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

    34 Hypertranscendental function 10634.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10634.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10634.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

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    34.3.1 Hypertranscendental functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10634.3.2 Transcendental but not hypertranscendental functions . . . . . . . . . . . . . . . . . . . . 10634.3.3 Non-transcendental (algebraic) functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

    34.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10734.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10734.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

    35 Identity function 10835.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10935.2 Algebraic property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10935.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10935.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10935.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

    36 Indicator function 11036.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11036.2 Remark on notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11136.3 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11136.4 Mean, variance and covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11236.5 Characteristic function in recursion theory, Gdels and Kleenes representing function . . . . . . . 11236.6 Characteristic function in fuzzy set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11236.7 Derivatives of the indicator function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11336.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11336.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11436.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

    37 Injective function 11537.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11637.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11737.3 Injections can be undone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12037.4 Injections may be made invertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12037.5 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12037.6 Proving that functions are injective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12137.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12137.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12237.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12237.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

    38 Integer-valued function 12338.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12438.2 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12438.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

    38.3.1 Graph theory and algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

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    38.3.2 Mathematical logic and computability theory . . . . . . . . . . . . . . . . . . . . . . . . 12438.3.3 Number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12438.3.4 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

    38.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

    39 Invex function 12639.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12639.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12639.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

    40 Koenigs function 12740.1 Existence and uniqueness of Koenigs function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

    40.1.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12740.2 Koenigs function of a semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12840.3 Structure of univalent semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12840.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12940.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

    41 Kostant partition function 13041.1 Relation to the Weyl character formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13041.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

    42 List of types of functions 13242.1 Relative to set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13242.2 Relative to an operator (c.q. a group or other structure) . . . . . . . . . . . . . . . . . . . . . . . . 13242.3 Relative to a topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13342.4 Relative to an ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13342.5 Relative to the real/complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13342.6 Ways of dening functions/Relation to Type Theory . . . . . . . . . . . . . . . . . . . . . . . . . 13342.7 Relation to Category Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13342.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

    43 Locally integrable function 13543.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

    43.1.1 Standard denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13543.1.2 An alternative denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13543.1.3 Generalization: locally p-integrable functions . . . . . . . . . . . . . . . . . . . . . . . . . 13643.1.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

    43.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13643.2.1 Lp, is a complete metric space for all p 1 . . . . . . . . . . . . . . . . . . . . . . . . . 13643.2.2 Lp is a subspace of L, for all p 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13743.2.3 L, is the space of densities of absolutely continuous measures . . . . . . . . . . . . . . . 137

    43.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

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    43.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13843.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13943.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13943.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14043.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

    44 Measurable function 14244.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14244.2 Caveat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14344.3 Special measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14344.4 Properties of measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14344.5 Non-measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14344.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14444.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14444.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

    45 Monotonic function 14545.1 Monotonicity in calculus and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

    45.1.1 Some basic applications and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14745.2 Monotonicity in functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14845.3 Monotonicity in order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14845.4 Monotonicity in the context of search algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 14945.5 Boolean functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14945.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14945.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14945.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14945.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

    46 Morphism of varieties 15146.1 Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15146.2 Ocial denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15146.3 Relation to rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15146.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15246.5 Fibers of a morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15246.6 Degree of a nite morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15346.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15346.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

    47 Negligible function 15447.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15447.2 Use in Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15547.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15547.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

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    48 Nowhere continuous function 15648.1 Dirichlet function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15648.2 Hyperreal characterisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15748.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15748.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15748.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

    49 Periodic function 15849.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15849.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15949.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15949.4 Double-periodic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15949.5 Complex example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16049.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

    49.6.1 Antiperiodic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16049.6.2 Bloch-periodic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16049.6.3 Quotient spaces as domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

    49.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16149.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16149.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

    50 Piecewise linear function 16350.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16350.2 Fitting to a curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16350.3 Fitting to data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16350.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16350.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16550.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16550.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

    51 Polyconvex function 16751.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

    52 Positive-denite function 16852.1 In dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16852.2 In analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

    52.2.1 Bochners theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16852.2.2 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

    52.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16952.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16952.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

    53 Positive-real function 170

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    53.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17053.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17153.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17153.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

    53.4.1 Irrational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17153.4.2 Matrix-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

    53.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

    54 Power function 17354.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

    55 Progressive function 176

    56 Proper convex function 17756.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17756.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

    57 Proto-value functions 17857.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17857.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17857.3 Basis functions from graph Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

    57.3.1 Graph construction on discrete state space . . . . . . . . . . . . . . . . . . . . . . . . . . 17957.3.2 Graph construction on continuous or large state space . . . . . . . . . . . . . . . . . . . . 179

    57.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17957.4.1 Least-squares approximation using proto-value functions . . . . . . . . . . . . . . . . . . 179

    57.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18057.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

    58 Pseudoanalytic function 18158.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18158.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

    59 Pseudoconvex function 18259.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18259.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18259.3 Generalization to nondierentiable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18259.4 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18359.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18359.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18359.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

    60 Quasiconvex function 18460.1 Denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18660.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

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    60.2.1 Mathematical optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18760.2.2 Economics and partial dierential equations: Minimax theorems . . . . . . . . . . . . . . . 188

    60.3 Preservation of quasiconvexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18860.3.1 Operations preserving quasiconvexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18860.3.2 Operations not preserving quasiconvexity . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

    60.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18860.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18860.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18960.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

    61 Quasiperiodic function 19061.1 Quasiperiodic signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19161.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19161.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

    62 Quasisymmetric function 19262.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19262.2 Important bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19262.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19362.4 Related algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19362.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19462.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

    63 Radial function 19663.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19663.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

    64 Radially unbounded function 19864.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

    65 Radonifying function 19965.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19965.2 Push forward of a CSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19965.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

    66 Real-valued function 20166.1 In general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20166.2 Measurable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20266.3 Continuous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20266.4 Smooth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20266.5 Appearances in measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20266.6 Other appearances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20366.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20366.8 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

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    66.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

    67 Regulated function 20467.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20467.2 Properties of regulated functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20467.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

    68 Representative function 20668.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

    69 Ring of symmetric functions 20769.1 Symmetric polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20769.2 The ring of symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

    69.2.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20869.2.2 Dening individual symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 20969.2.3 A principle relating symmetric polynomials and symmetric functions . . . . . . . . . . . . 210

    69.3 Properties of the ring of symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21069.3.1 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21069.3.2 Structural properties of R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21169.3.3 Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

    69.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21269.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

    70 Round function 21370.1 For instance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21370.2 Round complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21470.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

    71 Rvachev function 21571.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21571.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21571.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

    72 Simple function 21672.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21672.2 Properties of simple functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21672.3 Integration of simple functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21672.4 Relation to Lebesgue integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21772.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

    73 Single-valued function 21873.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21873.2 Injective functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

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    74 Singular function 22074.1 When referring to functions with a singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22174.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22174.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

    75 Slowly varying function 22275.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22275.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22275.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22375.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

    76 Subharmonic function 22476.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22476.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22476.3 Subharmonic functions in the complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

    76.3.1 Harmonic majorants of subharmonic functions . . . . . . . . . . . . . . . . . . . . . . . . 22576.3.2 Subharmonic functions in the unit disc. Radial maximal function . . . . . . . . . . . . . . 225

    76.4 Subharmonic functions on Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 22676.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22676.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22676.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

    77 Sublinear function 22877.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22877.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22877.3 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22877.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

    78 Support function 22978.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22978.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22978.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

    78.3.1 As a function of x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23078.3.2 As a function of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

    78.4 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23178.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23178.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

    79 Supporting functional 23279.1 Mathematical denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23279.2 Relation to support function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23279.3 Relation to supporting hyperplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23279.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

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    80 Surjective function 23380.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23480.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23480.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

    80.3.1 Surjections as right invertible functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23580.3.2 Surjections as epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23680.3.3 Surjections as binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23780.3.4 Cardinality of the domain of a surjection . . . . . . . . . . . . . . . . . . . . . . . . . . . 23780.3.5 Composition and decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23780.3.6 Induced surjection and induced bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

    80.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23780.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23880.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

    81 Symmetrically continuous function 23981.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

    82 Test functions for optimization 24082.1 Test functions for single-objective optimization problems . . . . . . . . . . . . . . . . . . . . . . . 24082.2 Test functions for multi-objective optimization problems . . . . . . . . . . . . . . . . . . . . . . . 24082.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24082.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

    83 Transcendental function 24283.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24283.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24283.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24283.4 Algebraic and transcendental functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24383.5 Transcendentally transcendental functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24383.6 Exceptional set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24383.7 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24483.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24483.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24483.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

    84 Transfer function 24684.1 LTI systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

    84.1.1 Direct derivation from dierential equations . . . . . . . . . . . . . . . . . . . . . . . . . 24784.2 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

    84.2.1 Common transfer function families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24884.3 Control engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24984.4 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24984.5 Non-linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

  • xvi CONTENTS

    84.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24984.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25084.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

    85 Unary function 25185.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25185.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

    86 Vector-valued function 25286.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25286.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25386.3 Derivative of a three-dimensional vector function . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

    86.3.1 Partial derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25486.3.2 Ordinary derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25486.3.3 Total derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25486.3.4 Reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25486.3.5 Derivative of a vector function with nonxed bases . . . . . . . . . . . . . . . . . . . . . . 25486.3.6 Derivative and vector multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

    86.4 Derivative of an n-dimensional vector function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25586.5 Innite-dimensional vector functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

    86.5.1 Functions with values in a Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . 25686.5.2 Other innite-dimensional vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

    86.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25686.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25786.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25786.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

    87 Weakly measurable function 25887.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25887.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25887.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25887.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

    88 Weierstrass function 26088.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26088.2 Hlder continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26188.3 Density of nowhere-dierentiable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26288.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26288.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26288.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26388.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

    89 Weight function 264

  • CONTENTS xvii

    89.1 Discrete weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26489.1.1 General denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26489.1.2 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26589.1.3 Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

    89.2 Continuous weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26589.2.1 General denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26589.2.2 Weighted volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26689.2.3 Weighted average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26689.2.4 Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

    89.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26689.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

    90 Window function 26790.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

    90.1.1 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26790.1.2 Filter design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27090.1.3 Symmetry and asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27190.1.4 Applications for which windows should not be used . . . . . . . . . . . . . . . . . . . . . 271

    90.2 A list of window functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27290.2.1 B-spline windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27290.2.2 Other polynomial windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27490.2.3 Generalized Hamming windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27590.2.4 Higher-order generalized cosine windows . . . . . . . . . . . . . . . . . . . . . . . . . . 27690.2.5 Power-of-cosine windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27990.2.6 Adjustable windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28090.2.7 Hybrid windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28890.2.8 Other windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

    90.3 Comparison of windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29190.4 Overlapping windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29290.5 Two-dimensional windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29290.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29290.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29290.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29290.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29590.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

    91 Zonal spherical function 29691.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29691.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29891.3 Gelfand pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29891.4 CartanHelgason theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29991.5 Harish-Chandras formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

  • xviii CONTENTS

    91.6 Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30191.7 Example: SL(2,C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30291.8 Complex case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30391.9 Example: SL(2,R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30591.10Further directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30691.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30691.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30791.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30891.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30991.15Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 310

    91.15.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31091.15.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31691.15.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

  • Chapter 1

    Algebraic function

    This article is about algebraic functions in calculus, mathematical analysis, and abstract algebra. For functions inelementary algebra, see function (mathematics).

    In mathematics, an algebraic function is a function that can be dened as the root of a polynomial equation. Quiteoften algebraic functions can be expressed using a nite number of terms, involving only the algebraic operationsaddition, subtraction, multiplication, division, and raising to a fractional power:

    f(x) = 1/x; f(x) =px; f(x) =

    p1 + x3

    x3/7 p7x1/3are typical examples.However, some algebraic functions cannot be expressed by such nite expressions (as proven by Galois and NielsAbel), as it is for example the case of the function dened by

    f(x)5 + f(x)4 + x = 0

    In more precise terms, an algebraic function of degree n in one variable x is a function y = f(x) that satises apolynomial equation

    an(x)yn + an1(x)yn1 + + a0(x) = 0

    where the coecients ai(x) are polynomial functions of x, with coecients belonging to a set S. Quite often, S = Q ,and one then talks about function algebraic overQ ", and the evaluation at a given rational value of such an algebraicfunction gives an algebraic number.A function which is not algebraic is called a transcendental function, as it is for example the case of exp(x); tan(x); ln(x);(x). A composition of transcendental functions can give an algebraic function: f(x) = cos(arcsin(x)) =

    p1 x2 .

    As an equation of degree n has n roots, a polynomial equation does not implicitly dene a single function, but nfunctions, sometimes also called branches. Consider for example the equation of the unit circle: y2 + x2 = 1: Thisdetermines y, except only up to an overall sign; accordingly, it has two branches: y = p1 x2:An algebraic function in m variables is similarly dened as a function y which solves a polynomial equation in m +1 variables:

    p(y; x1; x2; : : : ; xm) = 0:

    It is normally assumed that p should be an irreducible polynomial. The existence of an algebraic function is thenguaranteed by the implicit function theorem.Formally, an algebraic function in m variables over the eld K is an element of the algebraic closure of the eld ofrational functions K(x1,...,xm).

    1

  • 2 CHAPTER 1. ALGEBRAIC FUNCTION

    1.1 Algebraic functions in one variable

    1.1.1 Introduction and overviewThe informal denition of an algebraic function provides a number of clues about the properties of algebraic func-tions. To gain an intuitive understanding, it may be helpful to regard algebraic functions as functions which can beformed by the usual algebraic operations: addition, multiplication, division, and taking an nth root. Of course, this issomething of an oversimplication; because of casus irreducibilis (and more generally the fundamental theorem ofGalois theory), algebraic functions need not be expressible by radicals.First, note that any polynomial function y = p(x) is an algebraic function, since it is simply the solution y to theequation

    y p(x) = 0:

    More generally, any rational function y = p(x)q(x) is algebraic, being the solution to

    q(x)y p(x) = 0:

    Moreover, the nth root of any polynomial y = npp(x) is an algebraic function, solving the equation

    yn p(x) = 0:

    Surprisingly, the inverse function of an algebraic function is an algebraic function. For supposing that y is a solutionto

    an(x)yn + + a0(x) = 0;

    for each value of x, then x is also a solution of this equation for each value of y. Indeed, interchanging the roles of xand y and gathering terms,

    bm(y)xm + bm1(y)xm1 + + b0(y) = 0:

    Writing x as a function of y gives the inverse function, also an algebraic function.However, not every function has an inverse. For example, y = x2 fails the horizontal line test: it fails to be one-to-one.The inverse is the algebraic function x = py . Another way to understand this, is that the set of branches of thepolynomial equation dening our algebraic function is the graph of an algebraic curve.

    1.1.2 The role of complex numbersFrom an algebraic perspective, complex numbers enter quite naturally into the study of algebraic functions. Firstof all, by the fundamental theorem of algebra, the complex numbers are an algebraically closed eld. Hence anypolynomial relation p(y, x) = 0 is guaranteed to have at least one solution (and in general a number of solutions notexceeding the degree of p in x) for y at each point x, provided we allow y to assume complex as well as real values.Thus, problems to do with the domain of an algebraic function can safely be minimized.Furthermore, even if one is ultimately interested in real algebraic functions, there may be no means to express thefunction in terms of addition, multiplication, division and taking nth roots without resorting to complex numbers (seecasus irreducibilis). For example, consider the algebraic function determined by the equation

    y3 xy + 1 = 0:

  • 1.1. ALGEBRAIC FUNCTIONS IN ONE VARIABLE 3

    A graph of three branches of the algebraic function y, where y3 xy + 1 = 0, over the domain 3/22/3 < x < 50.

    Using the cubic formula, we get

    y = 2x3p108 + 12p81 12x3

    +3p108 + 12p81 12x3

    6:

    For x 33p4 ; the square root is real and the cubic root is thus well dened, providing the unique real root. On theother hand, for x > 33p4 ; the square root is not real, and one has to choose, for the square root, either non real-squareroot. Thus the cubic root has to be chosen among three non-real numbers. If the same choices are done in the twoterms of the formula, the three choices for the cubic root provide the three branches shown, in the accompanyingimage.It may be proven that there is no way to express this function in terms nth roots using real numbers only, even thoughthe resulting function is real-valued on the domain of the graph shown.On a more signicant theoretical level, using complex numbers allows one to use the powerful techniques of complexanalysis to discuss algebraic functions. In particular, the argument principle can be used to show that any algebraicfunction is in fact an analytic function, at least in the multiple-valued sense.Formally, let p(x, y) be a complex polynomial in the complex variables x and y. Suppose that x0 C is such that thepolynomial p(x0,y) of y has n distinct zeros. We shall show that the algebraic function is analytic in a neighborhoodof x0. Choose a system of n non-overlapping discs i containing each of these zeros. Then by the argument principle

    1

    2i

    I@i

    py(x0; y)

    p(x0; y)dy = 1:

    By continuity, this also holds for all x in a neighborhood of x0. In particular, p(x,y) has only one root in i, given bythe residue theorem:

    fi(x) =1

    2i

    I@i

    ypy(x; y)

    p(x; y)dy

    which is an analytic function.

  • 4 CHAPTER 1. ALGEBRAIC FUNCTION

    1.1.3 MonodromyNote that the foregoing proof of analyticity derived an expression for a system of n dierent function elements fi(x),provided that x is not a critical point of p(x, y). A critical point is a point where the number of distinct zeros is smallerthan the degree of p, and this occurs only where the highest degree term of p vanishes, and where the discriminantvanishes. Hence there are only nitely many such points c1, ..., cm.A close analysis of the properties of the function elements fi near the critical points can be used to show that themonodromy cover is ramied over the critical points (and possibly the point at innity). Thus the entire functionassociated to the fi has at worst algebraic poles and ordinary algebraic branchings over the critical points.Note that, away from the critical points, we have

    p(x; y) = an(x)(y f1(x))(y f2(x)) (y fn(x))since the fi are by denition the distinct zeros of p. The monodromy group acts by permuting the factors, and thusforms the monodromy representation of the Galois group of p. (The monodromy action on the universal coveringspace is related but dierent notion in the theory of Riemann surfaces.)

    1.2 HistoryThe ideas surrounding algebraic functions go back at least as far as Ren Descartes. The rst discussion of algebraicfunctions appears to have been in Edward Waring's 1794 An Essay on the Principles of Human Knowledge in whichhe writes:

    let a quantity denoting the ordinate, be an algebraic function of the abscissa x, by the common methodsof division and extraction of roots, reduce it into an innite series ascending or descending according tothe dimensions of x, and then nd the integral of each of the resulting terms.

    1.3 See also Algebraic expression Analytic function Complex function Elementary function Function (mathematics) Generalized function List of special functions and eponyms List of types of functions Polynomial Rational function Special functions Transcendental function

    1.4 References Ahlfors, Lars (1979). Complex Analysis. McGraw Hill. van der Waerden, B.L. (1931). Modern Algebra, Volume II. Springer.

  • 1.5. EXTERNAL LINKS 5

    1.5 External links Denition of Algebraic function in the Encyclopedia of Math Weisstein, Eric W., Algebraic Function, MathWorld. Algebraic Function at PlanetMath.org. Denition of Algebraic function in David J. Darling's Internet Encyclopedia of Science

  • Chapter 2

    Almost periodic function

    In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic towithin any desired level of accuracy, given suitably long, well-distributed almost-periods. The concept was rststudied by Harald Bohr and later generalized by Vyacheslav Stepanov, Hermann Weyl and Abram SamoilovitchBesicovitch, amongst others. There is also a notion of almost periodic functions on locally compact abelian groups,rst studied by John von Neumann.Almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, butnot exactly. An example would be a planetary system, with planets in orbits moving with periods that are notcommensurable (i.e., with a period vector that is not proportional to a vector of integers). A theorem of Kroneckerfrom diophantine approximation can be used to show that any particular conguration that occurs once, will recur towithin any specied accuracy: if we wait long enough we can observe the planets all return to within a second of arcto the positions they once were in.

    2.1 MotivationThere are several inequivalent denitions of almost periodic functions. The rst was given by Harald Bohr. Hisinterest was initially in nite Dirichlet series. In fact by truncating the series for the Riemann zeta function (s) tomake it nite, one gets nite sums of terms of the type

    e(+it) logn

    with s written as ( + it) the sum of its real part and imaginary part it. Fixing , so restricting attention to a singlevertical line in the complex plane, we can see this also as

    ne(logn)it:

    Taking a nite sum of such terms avoids diculties of analytic continuation to the region < 1. Here the 'frequencieslog n will not all be commensurable (they are as linearly independent over the rational numbers as the integers n aremultiplicatively independent which comes down to their prime factorizations).With this initial motivation to consider types of trigonometric polynomial with independent frequencies, mathematicalanalysis was applied to discuss the closure of this set of basic functions, in various norms.The theory was developed using other norms by Besicovitch, Stepanov, Weyl, von Neumann, Turing, Bochner andothers in the 1920s and 1930s.

    2.1.1 Uniform or Bohr or Bochner almost periodic functionsBohr (1925) dened the uniformly almost-periodic functions as the closure of the trigonometric polynomials withrespect to the uniform norm

    6

  • 2.1. MOTIVATION 7

    jjf jj1 = supxjf(x)j

    (on bounded functions f on R). In other words, a function f is uniformly almost periodic if for every > 0 there is anite linear combination of sine and cosine waves that is of distance less than from f with respect to the uniformnorm. Bohr proved that this denition was equivalent to the existence of a relatively dense set of almost-periods,for all > 0: that is, translations T() = T of the variable t making

    jf(t+ T ) f(t)j < ":An alternative denition due to Bochner (1926) is equivalent to that of Bohr and is relatively simple to state:

    A function f is almost periodic if every sequence {(t + Tn)} of translations of f has a subsequencethat converges uniformly for t in (, +).

    The Bohr almost periodic functions are essentially the same as continuous functions on the Bohr compactication ofthe reals.

    2.1.2 Stepanov almost periodic functionsThe space Sp of Stepanov almost periodic functions (for p 1) was introduced by V.V. Stepanov (1925). It containsthe space of Bohr almost periodic functions. It is the closure of the trigonometric polynomials under the norm

    jjf jjS;r;p = supx

    1

    r

    Z x+rx

    jf(s)jp ds1/p

    for any xed positive value of r; for dierent values of r these norms give the same topology and so the same spaceof almost periodic functions (though the norm on this space depends on the choice of r).

    2.1.3 Weyl almost periodic functionsThe space Wp of Weyl almost periodic functions (for p 1) was introduced by Weyl (1927). It contains the space Spof Stepanov almost periodic functions. It is the closure of the trigonometric polynomials under the seminorm

    jjf jjW;p = limr 7!1 jjf jjS;r;p

    Warning: there are nonzero functions with ||||W, = 0, such as any bounded function of compact support, so to geta Banach space one has to quotient out by these functions.

    2.1.4 Besicovitch almost periodic functionsThe space Bp of Besicovitch almost periodic functions was introduced by Besicovitch (1926). It is the closure of thetrigonometric polynomials under the seminorm

    jjf jjB;p = lim supx!1

    1

    2x

    Z xxjf(s)jp ds

    1/pWarning: there are nonzero functions with ||||B,p = 0, such as any bounded function of compact support, so to geta Banach space one has to quotient out by these functions.The Besicovitch almost periodic functions in B2 have an expansion (not necessarily convergent) as

  • 8 CHAPTER 2. ALMOST PERIODIC FUNCTION

    Xane

    int

    with an2 nite and n real. Conversely every such series is the expansion of some Besicovitch periodic function(which is not unique).The space Bp of Besicovitch almost periodic functions (for p 1) contains the space Wp of Weyl almost periodicfunctions. If one quotients out a subspace of null functions, it can be identied with the space of Lp functions onthe Bohr compactication of the reals.

    2.1.5 Almost periodic functions on a locally compact abelian groupWith these theoretical developments and the advent of abstract methods (the PeterWeyl theorem, Pontryagin dualityand Banach algebras) a general theory became possible. The general idea of almost-periodicity in relation to a locallycompact abelian group G becomes that of a function F in L(G), such that its translates by G form a relativelycompact set. Equivalently, the space of almost periodic functions is the norm closure of the nite linear combinationsof characters of G. If G is compact the almost periodic functions are the same as the continuous functions.The Bohr compactication ofG is the compact abelian group of all possibly discontinuous characters of the dual groupof G, and is a compact group containing G as a dense subgroup. The space of uniform almost periodic functions onG can be identied with the space of all continuous functions on the Bohr compactication of G. More generally theBohr compactication can be dened for any topological group G, and the spaces of continuous or Lp functions on theBohr compactication can be considered as almost periodic functions on G. For locally compact connected groups Gthe map from G to its Bohr compactication is injective if and only if G is a central extension of a compact group,or equivalently the product of a compact group and a nite-dimensional vector space.

    2.2 Quasiperiodic signals in audio and music synthesisIn speech processing, audio signal processing, and music synthesis, a quasiperiodic signal, sometimes called a quasi-harmonic signal, is a waveform that is virtually periodic microscopically, but not necessarily periodic macroscopi-cally. This does not give a quasiperiodic function in the sense of the Wikipedia article of that name, but somethingmore akin to an almost periodic function, being a nearly periodic function where any one period is virtually identicalto its adjacent periods but not necessarily similar to periods much farther away in time. This is the case for musi-cal tones (after the initial attack transient) where all partials or overtones are harmonic (that is all overtones are atfrequencies that are an integer multiple of a fundamental frequency of the tone).When a signal x(t) is fully periodic with period T , then the signal exactly satises

    x(t) = x(t+ T )

    or

    jx(t) x(t+ T )j = 0 all for t:The Fourier series representation would be

    x(t) =1

    2a0 +

    1Xn=1

    [an cos(2nf0t) bn sin(2nf0t)]

    or

    x(t) =1

    2a0 +

    1Xn=1

    [rn cos(2nf0t+ 'n)]

  • 2.2. QUASIPERIODIC SIGNALS IN AUDIO AND MUSIC SYNTHESIS 9

    where f0 = 1T is the fundamental frequency and the Fourier coecients are

    an = rn cos ('n) =2

    T

    Z t0+Tt0

    x(t) cos(2nf0t) dt

    bn = rn sin ('n) = 2T

    Z t0+Tt0

    x(t) sin(2nf0t) dt

    where t0 can be any time: 1 < t0 < +1 .

    The fundamental frequency f0 , and Fourier coecients an , bn , rn , or 'n , are constants, i.e. they are notfunctions of time. The harmonic frequencies are exact integer multiples of the fundamental frequency.When x(t) is quasiperiodic then

    x(t) x (t+ T (t))or

    jx(t) x (t+ T (t))j < "where

    0 < kxk =px2 =

    slim!1

    1

    Z /2/2

    x2(t) dt:

    Now the Fourier series representation would be

    x(t) =1

    2a0(t) +

    1Xn=1

    an(t) cos

    2n

    Z t0

    f0() d

    bn(t) sin

    2n

    Z t0

    f0() d

    or

    x(t) =1

    2a0(t) +

    1Xn=1

    rn(t) cos

    2n

    Z t0

    f0() d + 'n(t)

    or

    x(t) =1

    2a0(t) +

    1Xn=1

    rn(t) cos

    2

    Z t0

    fn() d + 'n(0)

    where f0(t) = 1T (t) is the possibly time-varying fundamental frequency and the Fourier coecients are

    an(t) = rn(t) cos ('n(t))

    bn(t) = rn(t) sin ('n(t))and the instantaneous frequency for each partial is

    fn(t) = nf0(t) +1

    2'0n(t):

  • 10 CHAPTER 2. ALMOST PERIODIC FUNCTION

    Whereas in this quasiperiodic case, the fundamental frequency f0(t) , the harmonic frequencies fn(t) , and theFourier coecients an(t) , bn(t) , rn(t) , or 'n(t) are not necessarily constant, and are functions of time albeitslowly varying functions of time. Stated dierently these functions of time are bandlimited to much less than thefundamental frequency for x(t) to be considered to be quasiperiodic.The partial frequencies fn(t) are very nearly harmonic but not necessarily exactly so. The time-derivative of 'n(t), that is '0n(t) , has the eect of detuning the partials from their exact integer harmonic value nf0(t) . A rapidlychanging 'n(t) means that the instantaneous frequency for that partial is severely detuned from the integer harmonicvalue which would mean that x(t) is not quasiperiodic.

    2.3 See also Quasiperiodic function Aperiodic function Quasiperiodic tiling Fourier series Additive synthesis Harmonic series (music) Computer music

    2.4 Notes

    2.5 References Amerio, Luigi; Prouse, Giovanni (1971), Almost-periodic functions and functional equations, The University

    Series in Higher Mathematics, New YorkCincinnatiTorontoLondonMelbourne: Van Nostrand Reinhold,pp. viii+184, ISBN 0-442-20295-4, MR 275061, Zbl 0215.15701.

    A.S. Besicovitch, On generalized almost periodic functions Proc. London Math. Soc. (2), 25 (1926) pp.495512

    A.S. Besicovitch, Almost periodic functions, Cambridge Univ. Press (1932) Bochner, S. (1926), Beitrage zur Theorie der fastperiodischen Funktionen, Math. Annalen 96: 119147,

    doi:10.1007/BF01209156

    S. Bochner and J. von Neumann, Almost Periodic Function in a Group II, Trans. Amer. Math. Soc., 37 no.1 (1935) pp. 2150

    H. Bohr, Zur Theorie der fastperiodischen Funktionen I Acta Math., 45 (1925) pp. 29127 H. Bohr, Almost-periodic functions, Chelsea, reprint (1947) Bredikhina, E.A. (2001), A/a011970, in Hazewinkel, Michiel, Encyclopedia ofMathematics, Springer, ISBN

    978-1-55608-010-4

    Bredikhina, E.A. (2001), Besicovitch almost periodic functions, in Hazewinkel, Michiel, Encyclopedia ofMathematics, Springer, ISBN 978-1-55608-010-4

    Bredikhina, E.A. (2001), Bohr almost periodic functions, in Hazewinkel, Michiel, Encyclopedia of Mathe-matics, Springer, ISBN 978-1-55608-010-4

    Bredikhina, E.A. (2001), Stepanov almost periodic functions, in Hazewinkel, Michiel, Encyclopedia ofMath-ematics, Springer, ISBN 978-1-55608-010-4

  • 2.6. EXTERNAL LINKS 11

    Bredikhina, E.A. (2001), Weyl almost periodic functions, in Hazewinkel, Michiel, Encyclopedia of Mathe-matics, Springer, ISBN 978-1-55608-010-4

    J. von Neumann, Almost Periodic Functions in a Group I, Trans. Amer. Math. Soc., 36 no. 3 (1934) pp.445492

    W. Stepano(=V.V. Stepanov), Sur quelques gnralisations des fonctions presque priodiques C.R. Acad.Sci. Paris, 181 (1925) pp. 9092

    W. Stepano(=V.V. Stepanov), Ueber einige Verallgemeinerungen der fastperiodischen Funktionen Math.Ann., 45 (1925) pp. 473498

    H. Weyl, Integralgleichungen und fastperiodische Funktionen Math. Ann., 97 (1927) pp. 338356

    2.6 External links Almost periodic function (equivalent denition) at PlanetMath.org.

  • Chapter 3

    Antiholomorphic function

    In mathematics, antiholomorphic functions (also called antianalytic functions) are a family of functions closelyrelated to but distinct from holomorphic functions.A function of the complex variable z dened on an open set in the complex plane is said to be antiholomorphic ifits derivative with respect to z exists in the neighbourhood of each and every point in that set, where z is the complexconjugate.One can show that if f(z) is a holomorphic function on an open set D, then f(z) is an antiholomorphic function on D,where D is the reection against the x-axis of D, or in other words, D is the set of complex conjugates of elementsof D. Moreover, any antiholomorphic function can be obtained in this manner from a holomorphic function. Thisimplies that a function is antiholomorphic if and only if it can be expanded in a power series in z in a neighborhoodof each point in its domain.If a function is both holomorphic and antiholomorphic, then it is constant on any connected component of its domain.

    12

  • Chapter 4

    Automorphic function

    In mathematics, an automorphic function is a function on a space that is invariant under the action of some group,in other words a function on the quotient space. Often the space is a complex manifold and the group is a discretegroup.

    4.1 Examples Kleinian group Elliptic modular function Modular function

    4.2 References Andrianov, A.N.; Parshin, A.N. (2001), Automorphic Function, in Hazewinkel, Michiel, Encyclopedia of

    Mathematics, Springer, ISBN 978-1-55608-010-4

    Ford, Lester R. (1929), Automorphic functions, New York, McGraw-Hill, ISBN 978-0-8218-3741-2, JFM55.0810.04

    Fricke, Robert; Klein, Felix (1897), Vorlesungen ber die Theorie der automorphen Functionen. Erster Band;Die grup