Complex Analysis: Theory and Applications · Problem 3.3.11

Sequences, Series, and Fractals — Howell & Matthews

Mathematics — Complex Analysis · polar-laplace-equation

Problem

Let f(z)=f(reiθ)=u(r,θ)+iv(r,θ)f(z) = f(re^{i\theta}) = u(r,\theta) + iv(r,\theta) be analytic on a domain DD that does not contain the origin. Use the polar form of the Cauchy-Riemann equations uθ=−rvru_\theta = -rv_r, and vθ=rurv_\theta = ru_r. Differentiate them first with respect to θ\theta and then with respect to rr. Use the results to establish the polar form of Laplace's equation: r2urr(r,θ)+rur(r,θ)+uθθ(r,θ)=0.r^2 u_{rr}(r,\theta) + ru_r(r,\theta) + u_{\theta\theta}(r,\theta) = 0.

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