From: lerma@math.nwu.edu (Miguel A. Lerma)
Subject: Re: ln |z| harmonnic?
Date: 29 May 2000 22:07:39 GMT
Newsgroups: sci.math
Summary: [missing]

Pansy (someone@somewhere.org) wrote:
: I'm reading through Francis Flanagan's "Complex Variables: Harmonic and
: Analytic Functions", a Dover publication.  Before moving on to complex
: numbers, the author first covers differential and integral calculus in the
: x-y plane.  The author defines a harmonic function u(x,y) as one for which
: 
:     u_xx(x,y) + u_yy(x,y) = 0
: 
: Notation: u_xx is the second partial derivative of u with respect to x; u_yy
: is the same for y.
: 
: My question deals with exercise 1(d), page 45: is
: 
:     ln |z|  (where z = (x,y) and hence |z| = sqrt(x^2 + y^2))
: 
: harmonic?  The domain is not specified, but I assume the domain to be all
: (x,y) for which the function is defined, i.e. everywhere but the origin.

This is the most typical non trivial example of real harmonic 
function.  In fact, for u(x,y) = ln(sqrt(x^2 + y^2)):

   u_xx(x,y) = (x^2+y^2)/(x^2+y^2)^2

   u_yy(x,y) = - (x^2+y^2)/(x^2+y^2)^2

hence u_xx(x,y) + u_yy(x,y) = 0.

: I say no, but the answer key in the back of the book says yes.
: 
: I can compute the first partials u_x and u_y either in x-y or in polar
: coordinates.  Either way, hese come out to be:
: 
:     u_x(x,y) = x/sqrt(x^2 + y^2)
:     u_y(x,y) = y/sqrt(x^2 + y^2)

It is 

      u_x(x,y) = x/(x^2 + y^2)

      u_y(x,y) = y/(x^2 + y^2)


Later you will see that the real harmonic functions are precisely
the real part (or the imaginary part) or analytic functions. 
In fact ln |z| is the real part of ln z = ln |z| + i arg(z).
If u, v are real and u + i v is analytic, then v is called 
"harmonic" conjugate of u. So arg(z) = arctan y/x is the harmonic 
conjugate of ln |z| (in an appropriate domain, of course).


Miguel A. Lerma