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JacobiAnger expansionFrom Wikipedia, the free encyclopedia
In mathematics, the JacobiAnger expansion(or JacobiAnger
identity) is an expansion of exponentials o
trigonometric functions in the basis of their harmonics. It is
useful in physics (for example, to convert between
plane waves and cylindrical waves), and in signal processing (to
describe FM signals). This identity is named
after the 19th-century mathematicians Carl Jacobi and Carl
Theodor Anger.
The most general identity is given by:[1][2]
and
where is the n-th Bessel function. Using the relation valid for
integer n, t
expansion becomes:[1][2]
The following real-valued variations are often useful as
well:[3]
Notes
^abColton & Kress (1998) p. 32.1.
^ abCuyt et al.(2008) p. 344.2.
^Abramowitz & Stegun (1965) p. 361, 9.1.4245
(http://www.math.sfu.ca/~cbm/aands/page_361.htm)3.
References
JacobiAnger expansion - Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/JacobiAng
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5/23/2018 JacobiAnger Expansion - Wikipedia, The Free
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Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 9"
(http://www.math.sfu.ca/~cbm/aands
/page_355.htm),Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical
Tables, New York: Dover, p. 355, ISBN 978-0486612720, MR 0167642
(//www.ams.org/mathscinet-
getitem?mr=0167642).
Colton, David; Kress, Rainer (1998),Inverse acoustic and
electromagnetic scattering theory, Applied
Mathematical Sciences 93(2nd ed.), ISBN 978-3-540-62838-5
Cuyt, Annie; Petersen, Vigdis; Verdonk, Brigitte; Waadeland,
Haakon; Jones, William B. (2008),
Handbook of continued f ractions for special functions,
Springer, ISBN 978-1-4020-6948-2
External links
Weisstein, Eric W. "JacobiAnger expansion"
(http://mathworld.wolfram.com/Jacobi-
AngerExpansion.html). MathWorld a Wolfram web resource.
Retrieved 2008-11-11.
Retrieved from
"http://en.wikipedia.org/w/index.php?title=JacobiAnger_expansion&oldid=588891142"
Categories: Special functions Mathematical identities
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