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Chapter 6 Harmonic Functions
The shortest route between two truths in the real domain passes through the complex domain.
―Jacques Hadamard (1865–1963)
We will now spend a short while on certain functions defined on subsets of the complex plane that are
real valued, namely those functions that are harmonic in some region. The main motivation for studying harmonic functions is that the partial differential equation they satisfy is very common in the physical sciences. Their definition briefly showed its face in
Chapter 2 , but we study them only now in more detail, since we have more machinery at our disposal. This machinery comes from
complex -valued functions, which are, nevertheless, intimately connected to harmonic functions.
