What is the function of a stele?

What is an example of a harmonic function u(x, y) that is not the real part of any analytic function?

• It is not hard to show that any analytic function f(z) can be written in the form of u(x, y) + i * v(x, y), where x and y are the real and imaginary parts of z, and u(x, y) and v(x, y) are real-valued harmonic functions. However, I am told that there are some harmonic functions that cannot appear as part of an analytic function in this manner. How can I find a function that cannot appear as the real part of an analytic function in this manner? How can I prove that it is not the real part of any analytic function?

• Answer:

  You can always find a harmonic conjugate locally but you may not be able to find one that is defined in all of the original functions domain. The standard example is u = ln (x^2+y^2). A harmonic conjugate is v = 2 arctan(y/x) notice u is harmonic for all (x,y) away from the origin but v is not (it's discontinuous across the y-axis). Thus there can't be an analytic function f(z), defined on all points except the origin, with real part u. Good Luck!