# harmonic v(x,y)

#### pasleycakes

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
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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