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Conformal Transformations
-------------------------
Conformal maps preserve both angles and the shapes of
infinitesimally small figures, but not necessarily their
size or curvature. Another way of saying this is that
the angles at an infinitesimal (local) scale are preserved
under the transformation, but not necessarily lengths.
For example,
Courtesy of Wikipedia
Consider 2 complex planes z and w such that w = f(z).
In other words, a mapping of the z plane to the w
plane.
dw/dz = (du + idv)/(dx + idy)
Differentiate w.r.t. x:
= ∂u/∂x + i∂v/∂x
Differentiate w.r.t. y:
= (1/i)∂u/∂y + ∂v/∂y
Therefore:
∂u/∂x = ∂v/∂y ... 1.
and,
i∂v/∂x = (1/i)∂u/∂y
Or,
∂u/∂y = -∂v/∂x ... 2.
1. and 2. are the CAUCHY-RIEMANN EQUATIONS. Functions
which satisfy the C-R equations are said to be ANALYTIC.
∂^{2}u/∂x^{2} = ∂^{2}v/∂x∂y
and,
∂^{2}u/∂y^{2} = -∂^{2}v/∂y∂x
Adding gives:
∂^{2}u/∂x^{2} + ∂^{2}u/∂y^{2} = 0
Likewise,
∂^{2}v/∂x^{2} + ∂^{2}v/∂y^{2} = 0
This is LAPLACE'S EQUATION. It can be interpreted
as follows. Consider:
Figure 1.
The first derivative w.r.t. τ is given by:
∂x^{μ}/∂τ = [x(2) - x(5)]/ε
The 2nd derivative w.r.t. τ is the difference of
first derivatives:
∴ ∂^{2}x^{μ}/∂τ^{2} = [x(2) - x(5)]/ε - [x(5) - x(4)]/ε
∂x^{μ}/∂σ = [x(1) - x(5)]/ε
∴ ∂^{2}x^{μ}/∂σ^{2} = [x(1) - x(5)]/ε - [x(5) - x(3)]/ε
Therefore, omitting ε
∂^{2}x^{μ}/∂τ^{2} + ∂^{2}x^{μ}/∂σ^{2} = [x(2) - x(5)] - [x(5) - x(4)]
+ [x(1) - x(5)] - [x(5) - x(3)]
_{ } = x(1) + x(2) + x(3) + x(4) - 4x(5)
Therefore,
_{ } x(5) = (x(1) + x(2) + x(3) + x(4))/4
x(5) is interpreted as the average of the field
at the corners.
If a function satisfies Laplace's equation in one
domain and is transformed via a conformal map to
another plane domain the transformation will also
satisfy Laplace's equation.
Angle Preservation
------------------
In polar coordinates (r = 1):
Δz = exp(iφ) and δz = exp(i(φ + θ)) so δz/Δz = exp(iθ)
Δw = exp(iβ) so δw = exp(i(β + θ')) so δw/Δw = exp(iθ')
Now δw/δz = Δw/Δz for an analytic function. Therefore,
Δz/δz = Δw/δw
Which implies that, θ = θ' so the angles are preserved.
Example:
Consider the transformation exp(z): z -> w
W = exp(z)
u + iv = exp(x + iy)
= exp(x)exp(iy)
= exp(x)(cos(y) + isin(y))
u = exp(x)cos(y) and v = exp(x)sin(y)
∂u/∂x = exp(x)cos(y) and ∂v/∂y = exp(x)cosy(y)
∂u/∂y = -exp(x)sin(y) and ∂v/∂x = exp(x)sin(y)
Also,
∂^{2}u/∂x^{2} = exp(x)cos(y)
∂^{2}u/∂y^{2} = -exp(x)cos(y)
Therefore,
∂^{2}u/∂x^{2} + ∂^{2}u/∂y^{2} = 0
Both the C-R and Laplace equations are satisfied and
therefore exp(z) is an analytic function. Note that
the inverse of an analytic function is also analytic
since angles also have to be preserved in the opposite
direction. Therefore, log(z) is also an analytic
function.