(Solution) - Let the function f z u r iv r -(2025 Original AI-Free Solution)

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Let the function f (z) = u(r, ?) + iv(r, ?) be analytic in a domain D that does not include the origin. Using the Cauchy-Riemann equations in polar coordinates (Sec. 23) and assuming continuity of partial derivatives, show that throughout D the function u(r, ?) satisfies the partial differential equation
r2urr (r, ?) + rur (r, ?) + u?? (r, ?) = 0,
which is the polar form of Laplace's equation. Show that the same is true of the function v(r, ?).