Complex Variables

Problem set: 2

1. Show that the real and imaginary parts of a twice-differentiable function f(z∗) satisfy Laplace’sequation. Show that f(z∗) is nowhere analytic unless it is constant.

2. Let f(z) be analytic in some domain. Show that f(z) is necessarily a constant if either thefunction f(z)∗ is analytic or f(z) assumes only pure imaginary values in the domain.

3. Consider the following complex potential

Ω(z) = − k

2π

1

z, k real

referred to as a “doublet”. Calculate the corresponding velocity potential, stream function,and velocity field. Sketch the stream function. The value of k is usually called the strengthof the doublet.

4. Given the complex analytic function Ω(z) = z2, show that the real part of Ω, φ(x, y) =Re Ω(z), satisfies Laplace’s equation, ∇2

x,yφ = 0. Let z = (1−w)/(1 +w), where w = u+ iv.Show that Φ(u, v) = Re Ω(w) satisfies Laplace’s equation ∇2

u,vΦ = 0.

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