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There are some theorems in harmonic function theory that resemble results in complex analysis, like:

  • Holomorphic functions and complex functions are analytic;
  • Cauchy's integral formula in complex analysis and the mean value theorem in harmonic function theory;
  • The principle of maximum and minimum that works for harmonic and holomophic functions.
  • The real and imaginary parts of a holomorphic function are harmonic;

These results suggest that there are connections between these two areas and I would like to ask: how can each of these theories be used to develop the other?

PS: I'm really sorry for my really bad English.

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closed as too localized by Andres Caicedo, Willie Wong, Yemon Choi, Pete L. Clark, David Hansen Jan 20 '11 at 2:58

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The real part of a holomorphic function is harmonic –  Anthony Quas Jan 15 '11 at 19:05
    
...and the imaginary part too –  Francesco Polizzi Jan 15 '11 at 19:08
    
I added this information to the question. Thanks! –  Jahnke Jan 15 '11 at 19:15
2  
I think that any book on complex analysis (or several complex variables) will discuss the connection at great length, so there is no need to write essays, long or otherwise. –  Igor Rivin Jan 15 '11 at 22:26
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See the paper Harmonic functions from a complex analysis viewpoint, by Sheldon Axler. (Google it to find an online version.) –  Charles Staats Jan 16 '11 at 0:51

1 Answer 1

The Cauchy Reimann Equations imply that every holomorphic function satisfies Laplace's Equation and is therefore its real and imaginary components are harmonic.

You can also take a harmonic function u and construct, up to a constant, its harmonic conjugate v so that u and v satisfy the Cauchy Reimann Equations. Thus u+iv is a holomorphic function.

These proofs can be found in just about any Complex Analysis book. My favorite is Complex Analysis by Lars Ahlfors.

So in a sense a harmonic function is just the real component of a holomorphic function.

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Do the words "simply connected" need to be mentioned somewhere? –  Yemon Choi Jan 19 '11 at 23:41
    
If I am not mistaken, the construction of the harmonic conjugate of a given harmonic function works locally, so simple connectedness only becomes an issue on a larger scale. –  drbobmeister Jan 20 '11 at 4:37

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