Question

If w = f(z) represents a conformal transformation of a domain D in the z-plane into a domain D' of the w-plane then f(z) is an analytic function of z in D'. ​

Answer 1

Answer:

Conformal mapping

There are many problems in physical applied mathematics, eg, fluid mechanics, electrostatics, elasticity theory, heat conduction etc, which require the solution of Laplace’s equation

∇2ϕ=0,

in some domain D with suitable boundary conditions.

Note that if P(z)=ϕ(x,y)+iψ(y) is an analytic function of the complex variable z=x+iy then from the Cauchy Riemann equations

ϕx=ψy,ϕy=−ψx

we obtain

∇2ϕ=ψxy−ψyx=0

and similarly

∇2ψ=0.

Thus the problem of solving Laplace’s equation can be reduced to finding an analytic function which satisfies certain boundary conditions.

In general if the domain D is complicated then this might have to done numerically. However by using a suitable mapping function w=f(z) the problem can be simplified if the domain can be transformed to the upper-half plane or the unit disk say. This is where conformal mapping is extremely useful

DefinitionA mapping is conformal if it preserves the angle between two differentiable arcs. A mapping defined by analytic functions is conformal.

Proof

Step-by-step explanation:

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