Conformal Mapping (6A) · 12/17/2013 · Conformal Mapping (6A) 12 Young Won Lim 12/17/13 Holomorphic and Analytic The term analytic function is often used interchangeably with “holomorphic

Young Won Lim12/17/13

Conformal Mapping (6A)


Copyright (c) 2013 Young W. Lim.

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Conformal Mapping (6A) 3 Young Won Lim12/17/13

Visualizing Functions of a complex variable

z = x + i y

ℜ

ℑ

ℜ

ℑ

ℜ

ℑ

ℜ

ℑ

ℜ

ℑ

ℜ

ℑ

f (z ) = u(x , y) + iv (u , y )

u(x , y )

v (x , y )


Complex Differentiable (z-differentiable )

f (z ) = ℜ(z)

d fdz

= limΔ→0

f (z+Δ z)− f (z )

Δ z

= limΔ→0

ℜ(z+Δ z )−ℜ(z)Δ z

= limΔ→0

ℜ(Δ z)

Δ z

Δ zreal

ℜ(Δ z ) = Δ zd fdz

=+1

Δ zimaginary

ℑ(Δ z) = 0d fdz

=+0

Depends on an approaching path

continuous everywheredifferentiable nowhere

f (z ) = z∗

d fdz

= limΔ→0

f (z+Δ z)− f (z )

Δ z

= limΔ→0

(z∗+Δ z∗

)−z∗

Δ z∗

=

limΔ→0

Δ z∗

Δ z=

∣a∣e−iθ

∣a∣e+iθ = e−i2θ

Δ zreal

θ =0d fdz

=+1

Δ zimaginary

θ = π/2d fdz

=−1

Depends on an approaching path

continuous everywheredifferentiable nowhere


Complex Differentiable

z-differentiablecomplex-differentiable

d fdz

= limΔ→0

f (z+Δ z)− f (z )

Δ z

f '(z0) = limz→ z0

f (z)− f (z0)

z−z0

z-derivativecomplex-derivative

the limist exists when it has the same value for any direction of approaching z

0

● linear operator● the product rule ● the quotient rule ● the chain rule

ℜ

ℑ

f(z)=z is not complex-differentiable at zero, because the value of (f(z) − f(0)) / (z − 0) is different depending on the approaching direction.

● along the real axis, the limit is 1,

● along the imaginary axis, the limit is −1.

● other directions yield yet other limits.


Cauchy-Rieman Equations

∂u∂ x

=∂v∂ y

∂u∂ y

=−∂v∂x

if f(z) is differentiable in a neighborhood of z(if df/dz exists)

∂u∂ x

=∂v∂ y

∂u∂ y

=−∂v∂x

conjugate functions:u(x,y), v(x,y) Re and Im part of a differentiable function f(z) if C-R eq holds

in a neighborhood of z = x + iy

The necessary condition

The sufficient condition

Cauchy-Rieman Equations: The necessary and sufficient condition


C-R Eq : the necessary condition

∂u∂ x

=∂v∂ y

∂u∂ y

=−∂v∂x

d fd z

= limΔz →0

f (z+Δ z) − f (z )

Δ z

Δ z = Δ x + i⋅0

ℜ(d fd z ) = limΔ x→0

u (x+Δ x , y ) − u(x , y )

Δ x

ℑ(d fd z ) = lim

Δ x→0

v (x+Δx , y) − v(x , y )

Δx

d fd z

= ℜ( d fd z ) + i⋅ℑ( d fd z ) =∂u∂ x

+ i⋅∂v∂ x

d fd z

= limΔz →0

f (z+Δ z) − f (z )

Δ z

Δ z =0 + i⋅Δ y

ℜ(d fdz ) = limΔ y→0

u(x , y+Δ y ) − u(x , y)

iΔ y

ℑ(d fdz ) = lim

Δ x→0

v (x , y+Δ y )− v(x , y)

iΔ y

d fd z

= ℜ( d fd z ) + i⋅ℑ(d fd z ) = −i∂u∂ y

+⋅∂v∂y

the necessary conditions on u(x,y) and v(x,y)if f(z) is differentiable in a neighborhood of z

if df/dz exists, then this C-R equations necessarily hold.


C-R Eq : the sufficient condition (1)

f (z+Δ z) = u (x+α , y+β) + iv (x+α , y+β)

= [u (x+α , y+β) −u(x , y)]

u(x+α , y+β) − u (x, y) = α∂u∂x

+ β∂u∂ y

+ o(Δ z )

v (x+α , y+β) − v (x , y ) = α∂v∂x

+β∂v∂y

+ o(Δ z )

− f (z ) − iv (x , y )=−u(x , y )

+ i[v (x+α , y+β) −v (x , y )]

o(a) : a remainder that goes to zero faster than a (little oh of a)

O(a) : a remainder that goes to zero as fast as a (big Oh of a)

= [α ∂u∂x

+ β∂u∂y

+ o(Δ z )] + i[α ∂v∂x

+ β∂v∂ y

+ o(Δ z )]= α[ ∂u∂x

+ i∂v∂x ] + β [ ∂u∂y

+ i∂v∂ y ] + o(Δ z )

f (z+Δ z) − f (z ) =f (z+Δ z) − f (z )

Δ z = α + iβ Δ x = α Δ y = β

d fd z

= limΔ z →0

f (z+Δ z ) − f (z )

Δ z

ℜ(d fdz ) = limΔ z→0

u (x+α , y+β) −u(x , y)

α + iβ

ℑ(d fdz ) = lim

Δ z→0

v (x+α , y+β) −v(x , y)

α + iβ


C-R Eq : the sufficient condition (2)

= α[ ∂u∂x+ i

∂v∂x ] + β [ ∂u∂y

+ i∂v∂ y ] + o(Δ z )

= α[ ∂v∂ y− i

∂u∂ y ] + iβ[−i ∂u∂ y

+∂v∂ y ] + o(Δz )

= [α + iβ ][ ∂v∂ y− i

∂u∂ y ] + o(Δ z )

d fd z

= limΔ z→0 [

α + iβα + iβ ][ ∂v∂ y

− i∂u∂ y ] = [ ∂v∂ y

− i∂u∂ y ]

f (z+Δ z) − f (z ) =

conjugate functions: u(x,y), v(x,y) Re and Im part of a differentiable function f(z)

the limit is independent of a path

= α[ ∂u∂ x+ i

∂v∂x ] + β [ ∂u∂y

+ i∂v∂ y ] + o(Δ z )

∂u∂x

=∂v∂ y

∂u∂ y

=−∂ v∂ x

= α[ ∂u∂ x+ i

∂v∂x ] + iβ [ ∂u∂x

+ i∂v∂ x ] + o(Δz )

= [α + iβ ][ ∂u∂ x+ i

∂v∂x ] + o(Δ z )

d fdz

= limΔ z→0 [

α + iβα + iβ ][ ∂u∂x

+ i∂v∂ x ] = [ ∂u∂ x

+ i∂v∂x ]

f (z+Δ z) − f (z ) =

∂u∂ x

=∂v∂ y

∂u∂ y

=−∂v∂x

differentiable∂u∂x

=∂v∂ y

∂u∂ y

=−∂ v∂ x

differentiable


A Holomorphic Function

from the Greek ὅλος (holos) meaning "entire", and μορφή (morphē) meaning "form" or "appearance".

a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.

sometimes referred to as regular functions or as conformal maps.

A holomorphic function whose domain is the whole complex plane is called an entire function.

The existence of a complex derivative in a neighborhood

infinitely differentiable and equal to its own Taylor series.

f(z) is Holomorphic

Cauchy's Formula

Complex Analytic Function


Holomorphic

not just differentiable at z0

but differentiable everywhere

within some neighborhood of z0

in the complex plane.

holomorphic at z0

complex differentiable at every point z0 in an open set U

holomorphic on U

not just complex differentiable at z0

but complex differentiable everywhere

within some neighborhood of z0

in the complex plane.

u and v have first partial derivatives with respect to x and y, and satisfy the Cauchy–Riemann equations:

f(z) is Holomorphic

∂u∂ x

=∂v∂ y

∂u∂ y

=−∂v∂x


Holomorphic and Analytic

The term analytic function is often used interchangeably with “holomorphic function”,

a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.

any function (real, complex, or of more general type) that can be written as a convergent power series in a neighborhood of every point in its domain.

analytic function

holomorphic functions

a function that is locally given by a convergent power series.

its Taylor series about x0 converges to the function in a neighborhood of every point x0 in its domain.

analytic

infinitely differentiable


Real & Complex Analytic Functions

smooth but not analytic

f (x) = 0 (x ≤ 0) f (x) = exp(−1 /x) (x > 0)

Taylor series not equal at x=0 and x>o

This cannot occur with complex differentiable functions

all holomorphic functions are analytic

one of the most dramatic differences between real-variable and complex-variable analysis.

infinitely differentiable, but

real function

its Taylor series about x0 converges to the function in a neighborhood of every point x0 in its domain.

analytic

infinitely differentiable


Complex Holomorphic & Analytic Functions

a complex-valued function ƒ of a complex variable z:

is said to be holomorphic at a point a if it is complex differentiable at every point within some open disk centered at a, and

is said to be analytic at a if in some open disk centered at a it can be expanded as a convergent power series

all holomorphic functions are analytic

complex holomorphic functions

complex analytic functions


Complex Analytic Functions

∂u∂ x

=∂v∂ y

∂u∂ y

=−∂v∂x

f(z) is differentiablein a neighborhood of z(if df/dz exists)

f(z) is Analytic (Holomorphic)


then f(z) is infinitely differentiablein that neighborhood.

Cauchy's Formula

f(z) can be expanded in a Taylor series

ℜ

ℑ

ℜ

ℑ

Generalized Cauchy's Formula


Laplace's Equation

∂u∂ x

=∂v∂ y

∂u∂ y

=−∂v∂x

∂2u

∂ x2 +∂2u

∂ y2 = 0 ∂2v

∂ x2 +∂2v

∂ y2 = 0

∂u∂ x

=∂v∂ y

∂u∂ y

=−∂v∂x

∂2u

∂ x2 =∂2v

∂ x∂ y∂2u

∂ y2 =−∂2v

∂x∂ y∂2u

∂ x∂ y=

∂2v

∂ y2

∂2u

∂ y2 =−∂2v

∂x∂ y

∂u∂ x

=∂v∂ y

∂u∂ y

=−∂v∂x


f(z) is Analytic (Holomorphic)


Isocontour

∇u = ( ∂u∂ x, ∂u

∂ y ) ∇ v = ( ∂v∂ x, ∂v

∂ y )∇u⋅∇ v = ( ∂u∂x

,∂u∂ y )⋅( ∂v∂ x

,∂v∂ y )

= ( ∂u∂ x∂v∂x

+∂u∂ y

∂v∂ y )

∂u∂x

=∂v∂ y

∂u∂ y

=−∂v∂x

∇u⋅∇ v = ( ∂u∂x∂v∂x

−∂v∂x

∂u∂x ) = 0

∇u⋅∇ v = 0 ∇u ⊥ ∇v

isocontour of v(x,y)

isocontour of u(x,y)

the gradient of u(x,y) :orthogonal to its isocontour

∇u = (∂u∂x,

∂u∂ y ) ∇ v = (∂v∂x

,∂v∂ y )

the gradient of v(x,y) :orthogonal to its isocontour

∇u ⊥ ∇ v

orthogonal isocontours

the isocontours of u(x,y) and v(x,y) are orthogonal to each other whenever they cross with each other


Angle Preserving Mappings

z = x + i y

ℜ

ℑ

ℜ

ℑ

f (z ) = u(x , y) + iv (u , y )

z w = f (z )

z

d z1

d z2

w = f (z )

θ θ

z+d z1

z+dz2

dw1

dw2

w+dw1 = f (z+z1)

w+dw2

d z1 = ∣dz1∣ejθ1

d z2 = ∣dz2∣ejθ2

d z2

d z1

=∣d z2∣∣d z1∣

e j (θ2−θ1)

f (z+z1) − f (z) = w+dw1 − w

f (z+z2) − f (z) = w+dw2 − w

f '(z) dz1 = dw1

f '(z) dz2 = dw2

arg (dw2

dw1) = arg (

d z2d z1 ) = θ

f (z+Δ z ) − f (z) ≈ f '(z )Δ z

all angles are preserved by the mapping w = f(z)except where f'(z) = 0 and dw

1 = dw

2 = 0 at first order.

0⋅d z1 = dw1 = 0

0⋅d z2 = dw2 = 0

dw1, dw

2 at first order

f (z+Δ z) − f (z) = f '(z)Δ z

dw1, dw

2 at first order

f'(z) = 0 and dw

1 = dw


arg (dw2

dw1) undefined

regardless of θ = arg (d z2

d z1)


Angle Preserving Mappings

d z2

d z1

= (d z2

d z1)2

= (∣z2∣

∣z1∣)2

ej2(θ2−θ1) = 2θ

f ' '( z )

2d z1

2 = dw1

f ' '( z )

2d z2

2 = dw2

all angles are preserved by the mapping w = f(z)except where f'(z) = 0 and dw

1 = dw

2 = 0

at first order.

f'(z) = 0 f''(z) ≠ 0 dw

1 and dw

2 at second order

d z2

d z1

= (d z2

d z1)3

= (∣z2∣

∣z1∣)3

ej3 (θ2−θ1) = 3θ

f (3)( z)

3!d z1

3 = dw1

f (3)( z)

3 !d z2

2= dw2

f'(z) = 0 f''(z) = 0 f(3)(z) ≠ 0 dw

1 and dw

2 at third order

z = r ei θ

z2 = r2ei2θ

z = r ei θ

z3 = r3ei3θ

at such points, angles are doubled

at such points, angles are trippled

f (z) = z2

f '(z ) = 2 zf ' '(z) = 2

f (z) = z3

f '(z) = 3 z2

f ' '(z) = 6zf (3)(z) = 6


Critical Points and Conformal Mapping

At a critical point, ∇u = 0, then ∇v = ∇⊥ u = 0the vectors do not define tangent directions

the orthogonality of the level curves does not necessarily hold at critical points.

the critical points of u = the critical points of v = the critical points of f (z)= points where its complex derivative vanishes: f ′(z) = 0.

0⋅d z1 = dw1 = 0

0⋅d z2 = dw2 = 0

f'(z) = 0 and dw

1 = dw


arg (dw2

dw1) undefined

regardless of θ = arg (d z2

d z1)

isocontour of v(x,y)

isocontour of u(x,y)

∇u ⊥ ∇ v

Except a critical point A type of a critical point

all angles are preserved by the mapping w = f(z)

f '(z) = 0


Conformal Condition

For every point z where f is holomorphic and f'(z) ≠ 0,the mapping z → w = f (z) is conformal, i.e., it preserves angles.

f '(z0) = 0 f '(z0) ≠ 0

conformal

f (z0) =w0

The phrase "holomorphic at a point z0" means not just differentiable at z0, but differentiable everywhere within some neighborhood of z0.

The existence of a complex derivative in a neighborhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal to its own Taylor series..

tangent vectors dz to each curve at z0

are transformed into vectors dw at w0 = f(z

0) which are

• magnified by factor |f ′(z0)|

• rotated through angle ψ0 = arg{f ′(z0)}

⇒ angles between curves remain the same (conformal mapping)


Critical Points

tangent vectors dz to each curve at z0

are transformed into vectors dw at w0 = f(z

0) which are

• magnified by factor |f ′(z0)|

• rotated through angle ψ0 = arg{f ′(z0)}

⇒ angles between curves remain the same (conformal mapping)

f '(z0) = 0

f ' '(z0) = 0 f ' '(z0) ≠ 0

f (3)(z0) = 0 f (3)

(z0) ≠ 0

f '(z0) ≠ 0

conformal

m=2 double

m=3tripple

f (z0) =w0

dw1 & dw

2

at 3rd order

dw1 & dw

2

at 2nd order

dw1 & dw

2

at 1st order


Harmonic

a harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rn) which satisfies Laplace's equation, i.e.

The solutions of Laplace's equation are the harmonic functions,

The general theory of solutions to Laplace's equation is known as potential theory.

Laplace's Equation on an annulus (inner radius r=2 and outer radius R=4)with Dirichlet Boundary Conditions: u(r=2)=0 and u(R=4)=4sin(5*θ)

∂2 f

∂ x12 +

∂2 f

∂ x22 + ⋯

∂2 f

∂ xn2 = 0

∇2 f = 0


Harmonic

In another direction, the study of oscillations of more general physical objects than strings leads to partial differential equations that are variations of the wave equation. Harmonic functions then encode the relative amplitudes at different places in the object for each mode of vibration.

why the name harmonic functions

fromhttp://math.stackexchange.com/questions/123620/why-are-harmonic-functions-called-harmonic-functions


Cauchy-Rieman Equation

∂u∂ x

=∂v∂ y

∂u∂ y

=−∂v∂x

u(x,y), v(x,y) continuous and differentiable and

f(z) is Holomorphic

f(z) is Holomorphic

f(z) is continuous

f(z) & g(z) areHolomorphic

c1f(z)+c

2g(z)

f(z)g(z)f(z)∘g(z)

arecontinuous

f(z) is Holomorphic& real value

f(z) is constant

f(z) is Holomorphic& |f(z)| const

f(z) is constant


Conformal Mapping

∂u∂ x

=∂v∂ y

∂u∂ y

=−∂v∂x


Riemann Surface

A one-dimensional complex manifold. can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or a couple of sheets glued together.


References

[1] http://en.wikipedia.org/[2] http://planetmath.org/[3] M.L. Boas, “Mathematical Methods in the Physical Sciences”[4] E. Kreyszig, “Advanced Engineering Mathematics”[5] D. G. Zill, W. S. Wright, “Advanced Engineering Mathematics”[6] T. J. Cavicchi, “Digital Signal Processing”[7] F. Waleffe, Math 321 Notes, UW 2012/12/11[8] J. Nearing, University of Miami[9] http://scipp.ucsc.edu/~haber/ph116A/ComplexFunBranchTheory.pdf[10] http://www.math.umn.edu/~olver/pd_/cm.pdf[11] http://wwwthphys.physics.ox.ac.uk/people/FrancescoHautmann/ComplexVariable/

s1_12_sl3p.pdf

http://scipp.ucsc.edu/~haber/ph116A/ComplexFunBranchTheory.pdf

http://www.math.umn.edu/~olver/pd_/cm.pdf

http://wwwthphys.physics.ox.ac.uk/people/FrancescoHautmann/ComplexVariable/

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Conformal Mapping (6A) · 12/17/2013 · Conformal Mapping (6A) 12 Young Won Lim 12/17/13 Holomorphic and Analytic The term analytic function is often used interchangeably with “holomorphic