harmonic v(x,y)

Dec 2008
5
0
Hi, can anyone help me prove the following.

Suppose that u(x,y) and v(x,y) are smooth functions satisfying the Cauchy-Riemann equations. Prove that v(x,y) is harmonic.

thank you
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Hi, can anyone help me prove the following.

Suppose that u(x,y) and v(x,y) are smooth functions satisfying the Cauchy-Riemann equations. Prove that v(x,y) is harmonic.

thank you
Yes, but the important question is, "Can you prove it?"
The Cauchy-Riemann equations are:
(1) \(\displaystyle \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}\)
(2) \(\displaystyle \frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}\).

Differentiate (1) with respect to y and (2) with respect to x.

Remember that, as long as the second derivatives are continuous,
\(\displaystyle \frac{\partial f}{\partial x\partial y}= \frac{\partial f}{\partial y\partial x}\).
 
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