Harmonic Functions

Steven G. Krantz
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Steven G. Krantz: Washington University in St. Louis, Department of Mathematics

Chapter Chapter 7 in Handbook of Complex Variables, 1999, pp 89-101 from Springer

Abstract: Abstract Let F be a holomorphic function on an open set $$U \subseteq \mathbb{C}$$ . Write F = u+iv, where u and v are real-valued. The real part u satisfies a certain partial differential equation known as Laplace’s equation: 7.1.1.1 $$\left( {\frac{{{{\partial }^{2}}}}{{\partial {{x}^{2}}}} + \frac{{{{\partial }^{2}}}}{{\partial {{y}^{2}}}}} \right)u = 0.$$ (Of course the imaginary part y satisfies the same equation.) In this chapter we shall study systematically those C2 functions that satisfy this equation. They are called harmonic functions. (Note that we encountered some of these ideas already in §1.4.)

Date: 1999
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DOI: 10.1007/978-1-4612-1588-2_7

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