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Question 5.25: Suppose f(z) = u(x, y) + iv(x, y) is analytic in a region R....

Suppose f(z) = u(x, y) + iv(x, y) is analytic in a region R. Prove that u and v are harmonic in R.

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In Problem 3.6, we proved that u and v are harmonic in R, i.e., satisfy the equation \left(\partial^2 \phi / \partial x^2\right)+\left(\partial^2 \phi / \partial y^2\right)=0, under the assumption of existence of the second partial derivatives of u and v, i.e., the existence of f″(z). This assumption is no longer necessary since we have in fact proved in Problem 5.4 that, if f(z) is analytic in R, then all the derivatives of f(z) exist

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