Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

a.) Show that if $v(x,y)$ is a harmonic conjugate of $u(x,y)$ in a domain $D$, then every harmonic conjugate of $u(x,y)$ in $D$ must be of the form $v(x,y)+a$, where $a$ is a real constant.

b.) Suppose that $f(z)$ is analytic and nonzero in a domain $D$. Prove that $ln|f(z)|$ is harmonic in $D$.

I know initutively why a is true, I also know that I must use Cauchy-Riemann equations to prove it but I do not know how to apply it correctly. For b, I am stumped.

share|improve this question

1 Answer 1

up vote 1 down vote accepted

(a)

Let $w$ be another such harmonic conjugate and so $z = u(x,y) + i w(x,y)$ where both $u$ and $w$ are real functions. $u_x = w_y$ so $u = \int w_y dx + C(y)$ where $C(y)$ is a pure real function of $y$.

Similarly, $u_y = -w_x$ so $u = -\int w_x dx + D(x)$ where $D(x)$ is a pure real function of $x$.

You know one possible antiderivative corresponding to $\int w_y dx$ and $-\int w_x dx$ already, namely $v(x,y)$ and so $C(y) = D(x)$. This is only possible when $C(y) = D(x) = a$, with $a$ a real constant.

(b)

$\ln f(z) = \ln|f(z)| + i \arg z$ and so it is the real part of an analytic function and so it is harmonic. I don't know if you're expected to do more than this

share|improve this answer
    
Does $u_x = w_y$ because of the Cauchy equations? And why does $u_y = v_x$ mean that $u = \int w_x dx + D(x)$, did you mean $u_y = w_x$ instead of $v_x$? –  Q.matin Mar 2 '13 at 6:36
    
Sorry, I posted it without really proof-reading it. Let me fix up a few details –  muzzlator Mar 2 '13 at 6:37
    
And yes, $u_x = w_y$ by the Cauchy-Riemann equations for $u$ being a harmonic conjugate of $w$. –  muzzlator Mar 2 '13 at 6:39
    
Thanks a lot Muzz! –  Q.matin Mar 2 '13 at 6:45
    
Oh and for part b it gives a hint saying, let $\phi (x,y)=ln|f(z)|=\frac{1}{2}ln(u^2 +v^2)$, will that be expected for an alternative proof? –  Q.matin Mar 2 '13 at 6:47

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.