u= ln(x2+y2) is a harmonic function

find a harmonic conjugate function v and the analytic function f(z) ?

Printable View

- January 16th 2010, 01:17 AMnice roseHarmonic conjugate function
u= ln(x2+y2) is a harmonic function

find a harmonic conjugate function v and the analytic function f(z) ?

- January 16th 2010, 02:00 AMshawsend
Try and be more precise with your notation. That's squares right? So first look at this thread:

http://www.mathhelpforum.com/math-he...xvariable.html

Then see if you can understand what I'm doing (lazy-style) with the following Mathematica code:

Code:`In[43]:=`

u[x_, y_] = Log[x^2 + y^2];

v[x_, y_] = FullSimplify[

Integrate[D[u[x, y], x], y] -

Integrate[D[Integrate[D[u[x, y], x],

y], x] + D[u[x, y], y], x]]

FullSimplify[D[u[x, y], x] -

D[v[x, y], y]]

FullSimplify[D[v[x, y], x] +

D[u[x, y], y]]

Out[44]=

2*ArcTan[y/x]

Out[45]=

0

Out[46]=

0

- January 16th 2010, 05:34 AMHallsofIvy
Any analytic function u(x,y)+ v(x,y)i must satisfy the "Cauchy-Riemann" equations: and [tex]\frac{\partial v}{\partial x}= -\frac{\partial u}{\partial y}.

The real and imaginary parts of any analytic function are harmonic functions and "v" is called the "harmonic conjugate" of u and vice versa.

Since and , the Cauchy-Riemann equations become and .

The integral so the substitution gives an "arctan" integral.