On the solution of Laplace’s equation in the vicinity of ...
Complex Variablesashwin/Mathematical_Physics...Complex Variables Problem set: 2 1.Show that the real...
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Complex Variables Problem set: 2 1. Show that the real and imaginary parts of a twice-differentiable function f (z * ) satisfy Laplace’s equation. 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 the function 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 strength of the doublet. 4. Given the complex analytic function Ω(z )= z 2 , show that the real part of Ω, φ(x, y)= Re Ω(z ), satisfies Laplace’s equation, ∇ 2 x,y φ = 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,v Φ = 0. 1
Transcript of Complex Variablesashwin/Mathematical_Physics...Complex Variables Problem set: 2 1.Show that the real...
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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