Back to the on-screen lesson ·
Differentiating the Cauchy-Riemann equations once more shows both parts of an analytic function satisfy Laplace's equation, which is the steady state of heat, electrostatics and ideal flow — and which brings the mean value property and the maximum principle with it.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to test a function of two real variables for harmonicity, prove that both parts of an analytic function are harmonic, recognise the harmonic functions that come from the standard analytic ones, find a coefficient that makes a given expression harmonic, and state the mean value property and the maximum principle together with what they are used for.
The Cauchy-Riemann equations, and the fact that an analytic function is infinitely differentiable — which will be proved in unit 3 and is assumed here, because it is what lets the two equations be differentiated a second time.
The Laplacian of $u$ is $u_{xx} + u_{yy}$, also written $\nabla^{2}u$. A function is harmonic on a region when it has continuous second partial derivatives there and its Laplacian is zero at every point of it — that is, when it satisfies Laplace's equation. The mean value property is the statement that the value at the centre of a circle is the average of the values on it.
The theorem. If $f = u + iv$ is analytic on a region, then both $u$ and $v$ are harmonic there.
The proof is two lines. Differentiate $u_x = v_y$ with respect to $x$ and $u_y = -v_x$ with respect to $y$:
$$u_{xx} = v_{yx}, \qquad u_{yy} = -v_{xy}.$$
The mixed partials of $v$ are equal, because $v$ has continuous second derivatives, so adding gives $u_{xx} + u_{yy} = 0$. The same computation the other way round gives $v_{xx} + v_{yy} = 0$.
Why this matters outside mathematics. Laplace's equation is the steady state of almost everything that diffuses: the temperature of a plate once it has settled, the electrostatic potential in a charge-free region, the velocity potential of an incompressible irrotational flow, the displacement of a stretched membrane. Every one of those is a harmonic function of two variables, and this theorem says every one of them is the real part of an analytic function — which is why complex analysis is a working tool in electrostatics and fluid dynamics rather than only a branch of pure mathematics.
What harmonic functions do. Three properties follow, and all three are proved in unit 3 from Cauchy's integral formula:
The last is why a boundary value problem has one answer rather than many, and the second is why a plate with its edges held at fixed temperatures never develops a hot spot in the middle.
And the converse, locally. Every harmonic function on a disc is the real part of some analytic function there. That is the next lesson: the missing imaginary part is built by integrating the Cauchy-Riemann equations.
Another way: picture
Think of $u$ as the height of a soap film stretched over a wire loop. Laplace's equation says the film has no slack anywhere: at every point its height is the average of the heights just around it. That is why there can be no bump in the middle — a bump would be higher than its own surroundings — and it is the maximum principle and the mean value property, both visible at once.
Another way: steps
There is an easy way to generate harmonic functions and it is worth knowing, because it turns the test into a recognition.
Take any function written in $z$ alone, expand it into real and imaginary parts, and both parts are harmonic. So:
| Analytic function | Real part | Imaginary part |
|---|---|---|
| $z$ | $x$ | $y$ |
| $z^{2}$ | $x^{2} - y^{2}$ | $2xy$ |
| $z^{3}$ | $x^{3} - 3xy^{2}$ | $3x^{2}y - y^{3}$ |
| $e^{z}$ | $e^{x}\cos y$ | $e^{x}\sin y$ |
| $\log z$ | $\tfrac12\ln(x^{2} + y^{2})$ | the argument |
Every entry in the last two columns is harmonic, and the last row is worth pausing on: $\ln|z|$ is harmonic on the punctured plane and is the potential of a point charge in two dimensions. It is also the standard example of a harmonic function with no single-valued harmonic conjugate on the punctured plane, because its conjugate is the argument — and that, again, is the ambiguity from unit 1, arriving in a new costume.
Going the other way, a function that is not harmonic cannot be the real part of anything analytic. $x^{2} + y^{2}$ has Laplacian $4$, so no analytic function has $|z|^{2}$ as its real part — which is a second, quicker proof of something the last lesson established by testing four partial derivatives.
The commonest mechanical error here is to compute $u_{xy}$ and put it into the test. It does not belong: the Laplacian is $u_{xx} + u_{yy}$, the two pure second derivatives, and the mixed one appears only in the proof of the theorem, where the two copies of it cancel.
The commonest conceptual error is to check the equation at a point and declare the function harmonic. Harmonic is a property of a region: $u = x^{3}$ has Laplacian $6x$, which vanishes on the imaginary axis, and $u$ is harmonic on no region whatever. A line is not a region, exactly as it was not in the last lesson.
A third slip is to expect the converse globally. Every harmonic function on a disc is the real part of an analytic function; on a region with a hole in it, that can fail, and $\ln|z|$ on the punctured plane is the example. The imaginary part it would need is the argument, which cannot be defined continuously all the way round.
$u = x^{3} - 3xy^{2}$: $u_{xx} = 6x$ and $u_{yy} = -6x$.
Two second derivatives.
They sum to zero at every point, so $u$ is harmonic — and expanding $z^{3}$ shows it is the real part of $z^{3}$.
Every harmonic function is somebody's real part.
$u = x^{2} + y^{2}$: $u_{xx} = 2$ and $u_{yy} = 2$, so the Laplacian is $4$.
Not zero anywhere.
So $u$ is harmonic on no region at all, and no analytic function has it as a real part. One second-derivative computation settled a question about every analytic function at once.
A negative result, cheaply.
$u_{xx} = e^{x}\sin y$ and $u_{yy} = -e^{x}\sin y$.
Differentiate twice in each variable.
They cancel, so yes — it is the imaginary part of the exponential.
Let $u = x^{3} - 3xy^{2}$. Fill in the table at the point $x = 4$, $y = 5$.
| Value | |
|---|---|
| The partial derivative in x | |
| The partial derivative in y | |
| The second partial derivative in x | |
| The second partial derivative in y | |
| The sum of the two second partial derivatives |
Is $u = x^{2} + y^{2}$ harmonic on the whole plane?
For which number $c$ is $u = 5x^{2} + cy^{2}$ harmonic?
Answer:
Put the steps of testing whether a function of two real variables is harmonic in order.
Number the steps in order (write the number in the box):
Match each harmonic function to an analytic function having it as its real part.
| $z^{2}$ | $z^{3}$ | $e^{z}$ | $7z$ | |
|---|---|---|---|---|
| $x^{2} - y^{2}$ | ||||
| $x^{3} - 3xy^{2}$ | ||||
| $e^{x}\cos y$ | ||||
| $7x$ |
$u$ is harmonic on a disc of radius $8$ and equals $7$ at every point of the boundary circle. What is $u$ at the centre?
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Let $u = x^{3} - 3xy^{2}$. Fill in the table at the point $x = 6$, $y = 5$.
| Value | |
|---|---|
| The partial derivative in x | |
| The partial derivative in y | |
| The second partial derivative in x | |
| The second partial derivative in y | |
| The sum of the two second partial derivatives |
You can test Laplace's equation and say why both parts of an analytic function pass it. Say in your own words why the mixed second derivative is not part of the test. Next: building the missing imaginary part, which turns the theorem around.
9. Your turn: is $u = e^{x}\sin y$ harmonic?, step 2