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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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  • $\begingroup$ Paragraph Connections with complex function theory on [en.wikipedia.org/wiki/Harmonic_function. $\endgroup$
    – Did
    Commented Jan 15, 2011 at 19:25
  • $\begingroup$ There is indeed a deep and well established connection, as other have pointed out. (In the other direction, given a harmonic function $f$, you can try to find a so called conjugate harmonic function $g$ so that $f+ig$ is holomorphic.) Unfortunately, it is hard to say much more without writing a long essay... $\endgroup$
    – Donu Arapura
    Commented Jan 15, 2011 at 21:45
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    $\begingroup$ 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. $\endgroup$
    – Igor Rivin
    Commented Jan 15, 2011 at 22:26
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    $\begingroup$ See the paper Harmonic functions from a complex analysis viewpoint, by Sheldon Axler. (Google it to find an online version.) $\endgroup$
    – Charles Staats
    Commented Jan 16, 2011 at 0:51
  • $\begingroup$ I would echo some of the comments above, and suggest that a book treating harmonic function theory in the plane (or pluriharmonic function theory in higher dimensions) would address the rather vague questions you raise. Ransford's *Potential Theory in the Complex Plane *is one example I happen to know; there ought to be others. $\endgroup$
    – Yemon Choi
    Commented Jan 16, 2011 at 4:28

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The Cauchy Riemann 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 Riemann 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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  • $\begingroup$ Do the words "simply connected" need to be mentioned somewhere? $\endgroup$
    – Yemon Choi
    Commented Jan 19, 2011 at 23:41
  • $\begingroup$ 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. $\endgroup$
    – drbobmeister
    Commented Jan 20, 2011 at 4:37

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