Let z = x + iy be a complex variable, and suppose that the function f : C → C given by f(z) = u(z) + iv(z) = u(x, y) + iv(x, y) is entire. Prove that h: C → R given by h(z) = h(x, y) = u^2(x, y) − v^2(x, y) is harmonic in C.

Let z = x + iy be a complex variable, and suppose that the function f : C → C given by f(z) = u(z) + iv(z) = u(x, y) + iv(x, y) is entire. Prove that h: C → R given by h(z) = h(x, y) = u^2(x, y) − v^2(x, y) is harmonic in C.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let z = x + iy be a complex variable, and suppose that the function
f : C → C given by f(z) = u(z) + iv(z) = u(x, y) + iv(x, y) is entire. Prove that h: C → R given by
h(z) = h(x, y) = u^2(x, y) − v^2(x, y) is harmonic in C.

Hint: Consider [f(z)]^2.

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