Complex Analysis

“ca: ch_02” — 2012/3/27 — 19:14 — page 86 — #46

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Complex Analysis

16. If v is harmonic conjugate of u, prove that:

(a) u is the harmonic conjugate of –v.

(b) –u is the harmonic conjugate of v.

(c) dv = u

x

dy − u

y

dx.

17. If f (z) = u(r, θ) + iv(r, θ) is an analytic function in a domain D that does not include the origin,

then by using the polar form of Cauchy–Riemann equations and assuming the continuity of partial

derivatives, show that throughout D the function u(r, θ) satisfies the partial differential equation

r

2

u

rr

(r, θ) + ru

r

(r, θ) + u

θθ

(r, θ) = 0 which is known as the polar form of Laplace’s equation.

Also, verify that the same is true in case ...