Harmonic Conjugate

**universalsandbox** · February 12th 2009, 01:28 PM

u = log|z|, Re(z) > 0

show that this is harmonic (i.e. satisfies laplace) and find the harmonic conjugate.

How would one do this? Thanks.

**HallsofIvy** · February 12th 2009, 05:33 PM

Originally Posted by universalsandbox

u = log|z|, Re(z) > 0

show that this is harmonic (i.e. satisfies laplace) and find the harmonic conjugate.

How would one do this? Thanks.

By, of course, using the Definitions of those things. Writing z= x+ iy, |z|= $\sqrt{x^2+ y^2}= (x^2+ y^2)^{1/2}$ so $u(x,y)= \log ((x^2+ y^2)^{1/2})= (1/2) \log(x^2+ y^2)$ .

u(x,y) is harmonic if and only if it satisfies $nabla^2 u= \frac{\partial^2 u}{\partial x^2}+ \frac{\partial^2 u}{\partial y^2}= 0$ . Find those partial derivatives and add!

If $f(z)= u(x,y)+ iv(x,y)$ is analytic then both u and v are harmonic functions and we say that they are "harmonic conjugates".

If f(z) is analytic it must satisfy $\frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}$ . Since you are given u, you can find the left sides of those and solve the resulting equations for v.

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