Chapter 7: Harmonic Functions

We have already seen that the real and imaginary parts of a holomorphic function are harmonic, as well as the rigidity in this formulation (the existence of harmonic conjugates for these parts).

This existence (if realized) of a conjugate is also unique. Indeed, on an open set, if there are two holomorphic functions ${𝑓}_1,{𝑓}_2$ with the same real part, then ${𝑓}_1-{𝑓}_2$ is a holomorphic function that is purely imaginary, and thus constant by Proposition 2.2.5$\text{<a href="../pdf/notes.pdf\#loc-55">Proposition 2.2.5</a>}\text{<a href="../introduction-to-complex-functions/complex-differentiation/index.html\#thm-proposition-2-2-5">Proposition 2.2.5</a>}$ (more generally by Theorem 5.1.1$\text{<a href="../pdf/notes.pdf\#loc-332">Theorem 5.1.1</a>}\text{<a href="../geometric-conformal-mappings/biholomorphy/index.html\#thm-theorem-5-1-1">Theorem 5.1.1</a>}$). Thus, the two holomorphic functions differ by a constant, and their imaginary parts differ by the same constant. Therefore, the harmonic conjugate of a harmonic function is unique up to an additive constant.

The most basic definition of a harmonic function requires ${𝐶}^2$. By a phenomenon known as elliptic regularity, harmonic functions are infinitely differentiable (in Theorem 7.1.2$\text{<a href="../pdf/notes.pdf\#loc-435">Theorem 7.1.2</a>}\text{<a href="properties-of-harmonic-functions/index.html\#thm-theorem-7-1-2">Theorem 7.1.2</a>}$); thus, the condition of ${𝐶}^2$ itself is not truly of great importance, except for establishing the initial definition.