ABSTRACT
This chapter discusses the basic properties of harmonic functions and real- and complex-valued harmonic functions. The definition of harmonic function applies well to complex-valued functions. A complex-valued function is harmonic if and only if its real and imaginary parts are each harmonic. The first thing that one needs to check is that real-valued harmonic functions are the functions that arise as the real parts of holomorphic functions—at least locally. Harmonic functions are provided as the real parts of holomorphic functions. An important and intuitively appealing consequence of the maximum principle is the “boundary maximum principle”. The Poisson integral formula is discussed along with the Poisson kernel. The Poisson integral formula both reproduces and creates harmonic functions.
