Question

Show that the function u=cosx coshy is a harmonic function.

Answer 1

Answer:

If a function f(z)=u(x,y)+iv(x,y), with z=x+iy and u,v:R2→R, is holomorphic, it is also harmonic. So, if you can find a v, such that the Cauchy-Riemann equations hold, i.e.: ∂u∂x=∂v∂y∂u∂y=−∂v∂x, you've found your harmonic conjugate

Answer 2

The function u = cosx coshy is a harmonic function.

To prove this function is a harmonic function we will prove that

$grad.^2$u(x,y) = 0

First, we will find the values of $\frac{d^2u}{dx^2}$ and $\frac{d^2u}{dy^2}$

u = cosx coshy

$\frac{du}{dx}$ = -sinx coshy

$\frac{d^2u}{dx^2}$ = - cosh coshy = -u

Similarly,

u = cosx coshy

$\frac{du}{dy}$ = cosx sinhy

$\frac{d^2u}{dy^2}$ = cosx coshy = u

Noe,

$grad.^2$u(x,y) = $\frac{d^2u}{dx^2}$ + $\frac{d^2u}{dy^2}$ = -u + u = 0

We can see that $grad.^2$u(x,y) = 0, therefore the given function is harmonic.