Thread: 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 $\displaystyle u_x = v_y \,\& \,u_y = - v_x $.
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 $\displaystyle f(x, y) = u(x, y) + i v(x, y)$ is analytic.

You want to prove that $\displaystyle g(x, y) = v(x, y) - i u(x, y)$ is analytic .......

Note that $\displaystyle g(x, y) = -i [i v(x, y) + u(x, y)] = -i f(x, y).$

Therefore .....

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Second:

So you have $\displaystyle f(x, y) = u(x, y) + i v(x, y)$ and $\displaystyle g(x, y) = v(x, y) + i u(x, y)$ are analytic.

You want to prove that u and v are constants.

From Cauchy-Riemann relations:

f analytic:

$\displaystyle \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \, $ and $\displaystyle \, \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$.


g analytic:

$\displaystyle \frac{\partial v}{\partial x} = \frac{\partial u}{\partial y} \, $ and $\displaystyle \, \frac{\partial v}{\partial y} = -\frac{\partial u}{\partial x}$.


Therefore:

$\displaystyle \frac{\partial u}{\partial x} = -\frac{\partial u}{\partial x} \, $ and $\displaystyle \frac{\partial u}{\partial y} = -\frac{\partial u}{\partial y} \, $.

It follows that u = constant.


$\displaystyle \frac{\partial v}{\partial x} = -\frac{\partial v}{\partial x} \, $ and $\displaystyle \frac{\partial v}{\partial y} = -\frac{\partial v}{\partial y} \, $.

It follows that v = constant.