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|= $\displaystyle \sqrt{x^2+ y^2}= (x^2+ y^2)^{1/2}$ so $\displaystyle u(x,y)= \log ((x^2+ y^2)^{1/2})= (1/2) \log(x^2+ y^2)$.
u(x,y) is harmonic if and only if it satisfies $\displaystyle nabla^2 u= \frac{\partial^2 u}{\partial x^2}+ \frac{\partial^2 u}{\partial y^2}= 0$. Find those partial derivatives and add!
If $\displaystyle f(z)= u(x,y)+ iv(x,y)$ 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 $\displaystyle \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}$ and $\displaystyle \frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}$. Since you are given u, you can find the left sides of those and solve the resulting equations for v.