I have two problems, but I am struggling to put the pieces together.
First:
Let v be a harmonic conjugate of u. Show that -u is a harmonic conjugate of v.
Second:
Suppose that v is a harmonic conjugate of u and that u is a harmonic conjugate of v. Show that u and v must be constant functions.
Can some push me in the right direction?
Thanks in advance
First:
So you have is analytic.
You want to prove that is analytic .......
Note that
Therefore .....
------------------------------------------------------------------------------------------------
Second:
So you have and are analytic.
You want to prove that u and v are constants.
From Cauchy-Riemann relations:
f analytic:
and .
g analytic:
and .
Therefore:
and .
It follows that u = constant.
and .
It follows that v = constant.