Originally Posted by
HallsofIvy What do you mean by "three main options in pairs"? I interpret this as asking which of these would be possible "u" parts of a function for which some "v" will give an analytic function.
I presume you know that if u(x,y)+ iv(x,y) is analytic if and only if u and v satisfy the "Cauchy-Riemann" equations:
$\displaystyle \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}$
$\displaystyle \frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}$
Now, notice what happens if we differentiate both sides of the first equation with respect to x and the second equation with respect to y:
$\displaystyle \frac{\partial^2 u}{\partial x^2}= \frac{\partial^2v}{\partial x\partial y}$
and
$\displaystyle \frac{\partial^2 u}{\partial y^2}= -\frac{\partial^2 v}{\partial x\partial y}$
That, is, u must satisfy
$\displaystyle \frac{\partial^2 u}{\partial x^2}= -\frac{\partial^2 u}{\partial y^2}$
or
$\displaystyle \frac{\partial^2 u}{\partial x^2}+ \frac{\partial^2 u}{\partial y^2}= 0$
That is, a function, u(x,y) can be part of an analytic if and only if it satisfies that equation.
(Surely whoever gave you this problem expects you to know that.)