4.8 Harmonic Functions
Let \(f(z) = u(x,y) + iv(x,y)\) and the second order partial differential equation \[\boxed {\frac {\partial ^2\Phi }{\partial x^2} + \frac {\partial ^2\Phi }{\partial y^2} = 0 \quad \cdots \quad (*)}\]
This equation is one of the most famous in applied mathematics. It is known as the Laplace’s equation
in two variables. The sum \(\frac {\partial ^2\Phi }{\partial x^2} + \frac {\partial ^2\Phi }{\partial y^2}\) of the two second partial derivatives in \((*)\) is denoted by \(\nabla ^2\Phi \) and is called the
Laplacian of \(\Phi \).
Definition 4.52. A real-valued function \(\Phi \) of two real variables \(x\) and \(y\) that has continuous first and second - order partial derivatives in a domain \(D\) and satisfies Laplace’s equation is said to be harmonic.
Theorem 4.53. Suppose the complex function \(f(z) = u(x,y) + iv(x,y)\) is analytic in a domain \(D\). Then the functions \(u(x,y)\) and \(v(x,y)\) are harmonic in \(D\).
Proof. An analytic function has derivatives of all orders, so \(u\) and \(v\) have continuous partial derivatives of every order and the mixed second partials may be taken in either order. Start from the Cauchy–Riemann equations \[u_x=v_y,\qquad u_y=-v_x .\] Differentiate the first with respect to \(x\) and the second with respect to \(y\): \[u_{xx}=v_{yx},\qquad u_{yy}=-v_{xy} .\] Adding, and using \(v_{xy}=v_{yx}\), \[u_{xx}+u_{yy}=v_{yx}-v_{xy}=0 .\] The same procedure with the roles reversed — differentiating \(u_x=v_y\) with respect to \(y\) and \(u_y=-v_x\) with respect to \(x\) — gives \[v_{xx}+v_{yy}=0 .\] So both \(u\) and \(v\) satisfy Laplace’s equation and are harmonic in \(D\). □
Remark 4.54. The converse is not automatic: a harmonic \(u\) is the real part of some analytic function only when the domain is simply connected. On an annulus \(\log \left |z\right |\) is harmonic but has no single-valued harmonic conjugate, which is the same obstruction that makes the logarithm multi-valued.
Example 4.55. The function \(f(z) = z^2 = x^2 - y^2 + 2xy i\) is entire. The functions \(u(x,y) = x^2 - y^2\) and \(v(x,y) = 2xy\) are necessary harmonic in any
domain on the complex plane.
We have just shown that if a function \(f(z) = u(x,y) + iv(x,y)\) is analytic in a domain \(D\), then its real and imaginary parts \(u\) and \(v\) are necessary harmonic in \(D\). However, suppose \(u(x,y)\) is a given real function that is known to be harmonic in \(D\). If it is possible to find another real harmonic function \(v(x,y)\) so that \(u\) and \(v\) satisfy the Cauchy-Riemann equations throughout \(D\), then the function \(v(x,y)\) is called a harmonic conjugate of \(u(x,y)\).
- 1.
- Verify that the function \(u(x,y) = x^3 - 3xy^2 - 5y\) is harmonic in the entire complex plane.
- 2.
- Find the harmonic conjugate function of \(u\)
Solution
From the partial derivatives
\[\frac {\partial u}{\partial x} = 3x^2 - 3y^2\quad , \qquad \frac {\partial ^2u}{\partial x^2} = 6x\]
\[\frac {\partial u}{\partial y} = -6xy - 5\quad ,\qquad \frac {\partial ^2u}{\partial y^2} = -6x\]
We see that the Laplace’s equation
\[\frac {\partial ^2u}{\partial x^2} + \frac {\partial ^2u}{\partial y^2} = 6x - 6x = 0\]
Since the conjugate harmonic function \(v\) must satisfy the Cauchy-Riemann equations
\[\frac {\partial v}{\partial y} = \frac {\partial u}{\partial x}\quad \text {and}\quad \frac {\partial v}{\partial x} = -\frac {\partial u}{\partial y}\]
We have that
\[\frac {\partial v}{\partial y} = 3x^2 -3y^2\quad \text {and}\quad \frac {\partial v}{\partial x}= 6xy + 5\]
Partial integration of the first equation with respect to \(y\) gives
\[v(x,y) = 3x^2y - y^3 + h(x)\quad \cdots \quad (*)\]
The partial derivative with respect to \(x\) of the last expression gives
\[\frac {\partial v}{\partial x} = 6xy + h'(x)\]
When this result is substituted into the second equation
\[6xy + h'(x) = 6xy + 5\]
We obtain \(h'(x) = 5\) and so \(h(x) = 5x + C\), where \(C\) is a constant. Therefore, the harmonic conjugate of \(h\) is
\[v(x,y) = 3x^2y - y^3 + 5x + C\]
And the function \(f(z) = x^3 - 3xy^2 + 5y + i (3x^2 - y^3 + 5x + C)\) is analytic throughout the domain \(D\), in this case the entire complex
plane.
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