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A First Course in Complex Analysis

Matthias Beck, Gerald Marchesi, Dennis Pixton, Lucas Sabalka

Section 6.1 Definition and Basic Properties

Recall from Section 2.3 the definition of a harmonic function:
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Definition 6.1.1.

Let \(G \subseteq \C\) be a region. A function \(u: G \to \R\) is harmonic in \(G\) if it has continuous second partials in \(G\) and satisfies the Laplace
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Named after Pierre-Simon Laplace (1749–1827).
equation
\begin{equation*} u_{xx} + u_{yy} \ = \ 0 \,\text{.} \end{equation*}
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Example 6.1.2.

The function \(u(x,y) = xy\) is harmonic in \(\C\) since \(u_{xx} + u_{yy} = 0 + 0 = 0\text{.}\)
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Example 6.1.3.

The function \(u(x,y) = e^x \cos(y)\) is harmonic in \(\C\) because
\begin{equation*} u_{xx} + u_{yy} \ = \ e^x \cos(y) - e^x \cos(y) \ = \ 0 \,\text{.} \end{equation*}
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There are (at least) two reasons why harmonic functions are part of the study of complex analysis, and they can be found in the next two results.
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Proof.

First, by Corollary 5.1.6, \(u\) and \(v\) have continuous second partials. By Theorem 2.3.1, \(u\) and \(v\) satisfy the Cauchy–Riemann equations (2.3)
\begin{equation*} u_x = v_y \qquad \text{ and } \qquad u_y = - v_x \end{equation*}
in \(G\text{.}\) Hence we can repeat our argumentation in (2.4),
\begin{equation*} u_{xx} + u_{yy} \ = \ \left( u_x \right)_x + \left( u_y \right)_y \ = \ \left( v_y \right)_x + \left( -v_x \right)_y \ = \ v_{yx} - v_{xy} \ = \ 0 \, \text{.} \end{equation*}
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Note that in the last step we used the fact that \(v\) has continuous second partials. The proof that \(v\) satisfies the Laplace equation is practically identical.
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Proposition 6.1.4 gives us an effective way to show that certain functions are harmonic in \(G\) by way of constructing an accompanying holomorphic function on \(G\text{.}\)
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Example 6.1.5.

Revisiting Example 6.1.2, we can see that \(u(x,y) = xy\) is harmonic in \(\C\) also by noticing that
\begin{equation*} f(z) \ = \ \tfrac 1 2 \, z^2 \ = \ \tfrac 1 2 \left( x^2 - y^2 \right) + i xy \end{equation*}
is entire and \(\Im(f) = u\text{.}\)
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Example 6.1.6.

A second reason that the function \(u(x,y) = e^x \cos(y)\) from Example 6.1.3 is harmonic in \(\C\) is that
\begin{equation*} f(z) \ = \ \exp(z) \ = \ e^x \cos(y) + i \, e^x \sin(y) \end{equation*}
is entire and \(\Re(f) = u\text{.}\)
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Proposition 6.1.4 practically shouts for a converse. There are, however, functions that are harmonic in a region \(G\) but not the real part (say) of a holomorphic function in \(G\) (Exercise 6.3.5). We do obtain a converse of Proposition 6.1.4 if we restrict ourselves to simply-connected regions.
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The function \(v\) is called a harmonic conjugate of \(u\text{.}\)
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Proof.

We will explicitly construct a holomorphic function \(f\) (and thus \(v = \Im f\)). First, let
\begin{equation*} g \ := \ u_x - i \, u_y \,\text{.} \end{equation*}
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The plan is to prove that \(g\) is holomorphic, and then to construct an antiderivative of \(g\text{,}\) which will be almost the function \(f\) that we’re after. To prove that \(g\) is holomorphic, we use Theorem 2.3.1: first because \(u\) is harmonic, \(\Re g = u_x\) and \(\Im g = - u_y\) have continuous partials. Moreover, again because \(u\) is harmonic, \(\Re g\) and \(\Im g\) satisfy the Cauchy–Riemann equations (2.3):
\begin{equation*} \left( \Re g \right)_x \ = \ u_{xx} \ = \ - u_{yy} \ = \ \left( \Im g \right)_y \end{equation*}
and
\begin{equation*} \left( \Re g \right)_y \ = \ u_{xy} \ = \ u_{yx} \ = \ - \left( \Im g \right)_x \, \text{.} \end{equation*}
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Theorem 2.3.1 implies that \(g\) is holomorphic in \(G\text{,}\) and so we can use Corollary 5.2.4 to obtain an antiderivative \(h\) of \(g\) on \(G\) (here is where we use the fact that \(G\) is simply connected). Now we decompose \(h\) into its real and imaginary parts as \(h=a+ib\text{.}\) Then, again using Theorem 2.3.1,
\begin{equation*} g \ = \ h' \ = \ a_x + i \, b_x \ = \ a_x - i \, a_y \,\text{.} \end{equation*}
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(The second equation follows from the Cauchy–Riemann equations (2.3).) But the real part of \(g\) is \(u_x\text{,}\) so we obtain \(u_x = a_x\) and thus \(u(x,y) = a(x,y) + c(y)\) for some function \(c\) that depends only on \(y\text{.}\) On the other hand, comparing the imaginary parts of \(g\) and \(h'\) yields \(-u_y = -a_y\) and so \(u(x,y) = a(x,y) + c(x)\) where \(c\) depends only on \(x\text{.}\) Hence \(c\) has to be constant, and \(u(x,y)=a(x,y)+c\text{.}\) But then
\begin{equation*} f(z) \ := \ h(z) + c \end{equation*}
is a function holomorphic in \(G\) whose real part is \(u\text{,}\) as promised.
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As a side remark, with hindsight it should not be surprising that the function \(g\) that we first constructed in our proof is the derivative of the sought-after function \(f\text{.}\) Namely, by Theorem 2.3.1 such a function \(f=u+iv\) must satisfy
\begin{equation*} f' \ = \ u_x + i \, v_x \ = \ u_x - i \, u_y \,\text{.} \end{equation*}
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(The second equation follows from the Cauchy–Riemann equations (2.3).) It is also worth mentioning that our proof of Theorem 6.1.7 shows that if \(u\) is harmonic in \(G\text{,}\) then \(u_x\) is the real part of the function \(g=u_x-iu_y\text{,}\) which is holomorphic in \(G\) regardless of whether \(G\) is simply connected or not.
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As our proof of Theorem 6.1.7 is constructive, we can use it to produce harmonic conjugates.
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Example 6.1.8.

Revisiting Example 6.1.2 for the second time, we can construct a harmonic conjugate of \(u(x,y) = xy\) along the lines of our proof of Theorem 6.1.7: first let
\begin{equation*} g \ := \ u_x - i \, u_y \ = \ y - i \, x \ = \ -i \, z \end{equation*}
which has antiderivative
\begin{equation*} h(z) \ = \ - \tfrac i 2 \, z^2 \ = \ xy - \tfrac i 2 \left( x^2 - y^2 \right) \end{equation*}
whose real part is \(u\) and whose imaginary part
\begin{equation*} v(x,y) \ := \ - \tfrac 1 2 \left( x^2 - y^2 \right) \end{equation*}
gives a harmonic conjugate for \(u\text{.}\)
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We can give a more practical machinery for computing harmonic conjugates, which only depends on computing certain (calculus) integrals; thus this can be easily applied, e.g., to polynomials. We state it for functions that are harmonic in the whole complex plane, but you can easily adjust it to functions that are harmonic on certain subsets of \(\C\text{.}\)
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Theorem 6.1.9 is due to Sheldon Axler and the basis for his Mathematica package Harmonic Function Theory.
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Proof.

We will prove that \(u+iv\) satisfies the Cauchy–Riemann equations  (2.3). The first follows from
\begin{equation*} \frac{ \partial v }{ \partial y } (x,y) = \frac{ \partial u }{ \partial x } (x,y) \, \text{,} \end{equation*}
by the Fundamental Theorem of Calculus (Theorem A.0.3).
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Second, by Leibniz’s Rule (Theorem A.0.9), the Fundamental Theorem of Calculus (Theorem A.0.3), and the fact that \(u\) is harmonic,
\begin{align*} \frac{ \partial v }{ \partial x } (x,y) \amp = \int_0^y \frac{ \partial^2 u }{ \partial x^2 } (x,t) \, \diff{t} - \frac{ \partial u }{ \partial y } (x,0)\\ \amp = - \int_0^y \frac{ \partial^2 u }{ \partial t^2 } (x,t) \, \diff{t} - \frac{ \partial u }{ \partial y } (x,0)\\ \amp = - \left( \frac{ \partial u }{ \partial y } (x,y) - \frac{ \partial u }{ \partial y } (x,0) \right) - \frac{ \partial u }{ \partial y } (x,0)\\ \amp = - \frac{ \partial u }{ \partial y } (x,y) \, \text{.} \end{align*}
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As you might imagine, Proposition 6.1.4 and Theorem 6.1.7 allow for a powerful interplay between harmonic and holomorphic functions. In that spirit, the following theorem appears not too surprising. You might appreciate its depth better when looking back at the simple definition of a harmonic function.
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Proof.

Suppose \(u\) is harmonic in \(G\) and \(z_0 \in G\text{.}\) We will show that \(u^{ (n) } (z_0)\) exists for all positive integers \(n\text{.}\) Let \(r>0\) such that the disk \(D[z_0,r]\) is contained in \(G\text{.}\) Since \(D[z_0,r]\) is simply connected, Theorem 6.1.7 asserts the existence of a holomorphic function \(f\) in \(D[z_0,r]\) such that \(u = \Re f\) on \(D[z_0,r]\text{.}\) By Corollary 5.1.6, \(f\) is infinitely differentiable on \(D[z_0,r]\text{,}\) and hence so is its real part \(u\text{.}\)
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This proof is the first in a series of proofs that uses the fact that the property of being harmonic is local—it is a property at each point of a certain region. Note that in our proof of Corollary 6.1.10 we did not construct a function \(f\) that is holomorphic in \(G\text{;}\) we only constructed such a function on the disk \(D[z_0,r]\text{.}\) This \(f\) might very well differ from one disk to the next.
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