Question 3.6: Given f(z) = u + iv is analytic in a region R. Prove that u ...
Given f(z) = u + iv is analytic in a region R. Prove that u and v are harmonic in R if they have continuous second partial derivatives in R.
If f(z) is analytic in R, then the Cauchy–Riemann equations
\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} (1)
and
\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y} (2)
are satisfied in R. Assuming u and v have continuous second partial derivatives, we can differentiate both sides of (1) with respect to x and (2) with respect to y to obtain
\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 v}{\partial x \partial y} (3)
and
\frac{\partial^2 v}{\partial y \partial x}=-\frac{\partial^2 u}{\partial y^2} (4)
from which
\frac{\partial^2 u}{\partial x^2}=-\frac{\partial^2 u}{\partial y^2} \quad \text { or } \quad \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0i.e., u is harmonic.
Similarly, by differentiating both sides of (1) with respect to y and (2) with respect to x, we find
\frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 v}{\partial y^2}=0and v is harmonic.
It will be shown later (Chapter 5) that if f(z) is analytic in R, all its derivatives exist and are continuous in R. Hence, the above assumptions will not be necessary.