Abstract
This chapter discusses Gauss's mean-value theorem, which follows directly from the Cauchy integral formula. It also discusses whether the integral formula yields a harmonic function despite of the discontinuities. This leads to the theorem called Poisson's integral formula. In addition, the chapter presents three analogous examples of steady state vector fields occurring in nature: fluid flow, heat flow, and the electrostatic field; these problems are extended to include sources and vortices in the domain. The theory is developed for fluid flow, and the analogies with the other two fields. The chapter also presents Maximum Principle that states that if u(z) is harmonic and nonconstant in a simply connected domain G, then u(z) has no maximum or minimum in G.
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