Math Help - Harmonic Conjugates

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

Originally Posted by ginafara

Let v be a harmonic conjugate of u. Show that -u is a harmonic conjugate of v.

Saying the v be a harmonic conjugate of u means that the function f=u+iv satisfies the Cauchy-Riemann equations.
Thus we know that .
If g=v+i(-u) does g satisfy the Cauchy-Riemann equations?

Originally Posted by ginafara

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 .....

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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.