Harmonic Functions in Complex Analysis

students, imagine a lake with no waves, no hills, and no dips—just a perfectly smooth surface 🌊. In Complex Analysis, a harmonic function is a mathematical version of that kind of smoothness. These functions appear whenever we study the Cauchy-Riemann equations, and they help connect complex functions to real-world ideas like temperature, gravity, electric potential, and fluid flow.

What you will learn

By the end of this lesson, students, you should be able to:

explain what a harmonic function is and why the term matters,
connect harmonic functions to the Cauchy-Riemann equations,
recognize examples and non-examples,
use derivatives to test whether a function is harmonic,
understand why harmonic functions are important in Complex Analysis.

1. What is a harmonic function?

A real-valued function of two variables, $u(x,y)$, is called harmonic if it satisfies Laplace’s equation:

$$u_{xx}+u_{yy}=0.$$

Here, $u_{xx}$ means the second partial derivative of $u$ with respect to $x$, and $u_{yy}$ means the second partial derivative with respect to $y$.

This equation says something very important: the function is balanced. At each point, the curvature in the $x$ direction cancels the curvature in the $y$ direction. That is why harmonic functions often model steady states, like a stable temperature distribution in a flat metal plate 🔥.

A harmonic function does not have local bumps or pits in the same way a hill-shaped surface does. Instead, it behaves smoothly and evenly. This is one reason harmonic functions are central in physics and engineering.

Example 1

Let $u(x,y)=x^2-y^2$.

Compute the second partial derivatives:

$$u_x=2x, \qquad u_{xx}=2,$$

$$u_y=-2y, \qquad u_{yy}=-2.$$

Now add them:

$$u_{xx}+u_{yy}=2+(-2)=0.$$

So $u(x,y)=x^2-y^2$ is harmonic ✅.

Example 2

Let $u(x,y)=x^2+y^2$.

Then

$$u_{xx}=2, \qquad u_{yy}=2,$$

$$u_{xx}+u_{yy}=4\neq 0.$$

So this function is not harmonic ❌.

2. Why harmonic functions matter in Complex Analysis

Harmonic functions become especially important when they are part of a complex-valued function $f(z)$, where

$$f(z)=u(x,y)+iv(x,y),$$

and $z=x+iy$.

If $f$ is complex differentiable, then its real part $u$ and imaginary part $v$ are closely linked by the Cauchy-Riemann equations:

$$u_x=v_y, \qquad u_y=-v_x.$$

A key fact in Complex Analysis is this:

if $f$ is analytic, then both $u$ and $v$ are harmonic.

This is a major bridge between complex functions and real-variable calculus. In simple terms, smooth complex functions produce two harmonic functions, one real part and one imaginary part.

Why does this happen?

If $u$ and $v$ satisfy the Cauchy-Riemann equations and the needed second derivatives exist and are continuous, then we can differentiate again:

From $u_x=v_y$, differentiate with respect to $x$:

$$u_{xx}=v_{yx}.$$

From $u_y=-v_x$, differentiate with respect to $y$:

$$u_{yy}=-v_{xy}.$$

Since mixed partial derivatives are equal when the functions are smooth enough,

$$v_{yx}=v_{xy},$$

so adding the two equations gives

$$u_{xx}+u_{yy}=0.$$

Thus $u$ is harmonic.

A similar calculation shows $v$ is harmonic too.

This means harmonic functions are not random side characters in Complex Analysis—they are direct consequences of analyticity ✨.

3. Working with examples from analytic functions

One of the best ways to understand harmonic functions is to see them come from familiar complex functions.

Example 3: $f(z)=z^2$

Write

$$z=x+iy.$$

Then

$$f(z)=(x+iy)^2=x^2-y^2+i(2xy).$$

So the real part is

$$u(x,y)=x^2-y^2,$$

and the imaginary part is

$$v(x,y)=2xy.$$

We already checked that $u$ is harmonic. Let’s test $v$:

$$v_x=2y, \qquad v_{xx}=0,$$

$$v_y=2x, \qquad v_{yy}=0.$$

Thus

$$v_{xx}+v_{yy}=0+0=0.$$

So $v$ is harmonic too.

Example 4: $f(z)=e^z$

Since

$$e^{x+iy}=e^x(\cos y+i\sin y),$$

we get

$$u(x,y)=e^x\cos y, \qquad v(x,y)=e^x\sin y.$$

For $u$:

$$u_{xx}=e^x\cos y, \qquad u_{yy}=-e^x\cos y,$$

$$u_{xx}+u_{yy}=0.$$

For $v$:

$$v_{xx}=e^x\sin y, \qquad v_{yy}=-e^x\sin y,$$

$$v_{xx}+v_{yy}=0.$$

Again, both parts are harmonic.

These examples show a pattern: analytic functions generate harmonic real and imaginary parts.

4. Harmonic functions and the “average value” idea

A powerful interpretation of harmonic functions is the average value property. Informally, the value of a harmonic function at a point is equal to the average of its values around that point, over a small circle or disk.

This idea helps explain why harmonic functions are so smooth. They do not have sudden spikes, because each point is “balanced” by nearby values.

For students, students, this can be thought of like a calm crowd where no one person suddenly shouts above everyone else—the local value reflects the surrounding neighborhood.

This property is one reason harmonic functions appear in models of:

steady heat distribution,
electrostatic potential,
incompressible fluid flow.

In each case, the system has reached a stable state, so the function describing it satisfies Laplace’s equation.

5. How to test whether a function is harmonic

To decide whether a function $u(x,y)$ is harmonic, follow these steps:

Compute $u_{xx}$.
Compute $u_{yy}$.
Add them.
Check whether the result is $0$.

Example 5

Let

$$u(x,y)=x^3-3xy^2.$$

First derivatives:

$$u_x=3x^2-3y^2, \qquad u_y=-6xy.$$

Second derivatives:

$$u_{xx}=6x, \qquad u_{yy}=-6x.$$

Add them:

$$u_{xx}+u_{yy}=6x+(-6x)=0.$$

So $u$ is harmonic.

This function is especially interesting because it is the real part of $z^3$:

$$z^3=(x+iy)^3=x^3-3xy^2+i(3x^2y-y^3).$$

So the real and imaginary parts are both harmonic, as expected.

6. Counterexamples and common mistakes

Not every smooth-looking function is harmonic. A common mistake is to think that any polynomial or any nice function must satisfy Laplace’s equation. That is false.

Counterexample 1

Let

$$u(x,y)=x^2+y^2.$$

As shown earlier,

$$u_{xx}+u_{yy}=4,$$

so it is not harmonic.

Counterexample 2

Let

$$u(x,y)=\sin x+\sin y.$$

Then

$$u_{xx}=-\sin x, \qquad u_{yy}=-\sin y,$$

$$u_{xx}+u_{yy}=-(\sin x+\sin y).$$

This is not identically zero, so the function is not harmonic.

A helpful way to avoid mistakes is to remember that harmonicity is not about being “simple” or “pretty.” It is about satisfying the exact equation

$$u_{xx}+u_{yy}=0.$$

7. Connection back to the Cauchy-Riemann equations

Harmonic functions are deeply tied to the Cauchy-Riemann equations because analytic functions require both structure and balance.

If $f(z)=u+iv$ is analytic and smooth enough, then:

$u$ is harmonic,
$v$ is harmonic,
$u$ and $v$ satisfy the Cauchy-Riemann equations.

So the Cauchy-Riemann equations are like the “bridge rules,” and harmonic functions are what you get on both sides of the bridge.

There is also an important partial converse: if one of the functions $u$ or $v$ is harmonic on a suitable region, it may be possible to find a partner function so that together they form an analytic function. That partner is called a harmonic conjugate. For example, if $u=x^2-y^2$, then a harmonic conjugate is $v=2xy$, giving the analytic function

$$f(z)=z^2.$$

This is a major idea in Complex Analysis because it shows that harmonic functions are not just outcomes—they can also help build complex functions.

Conclusion

Harmonic functions are smooth real-valued functions that satisfy Laplace’s equation

$$u_{xx}+u_{yy}=0.$$

They are important because they naturally appear as the real and imaginary parts of analytic functions. Through the Cauchy-Riemann equations, Complex Analysis shows that analytic behavior in the complex plane creates harmonic structure in the real plane.

students, the key takeaways are: harmonic functions are balanced, they are tested by second derivatives, and they connect directly to analytic complex functions. This makes them a central topic in understanding how Complex Analysis unifies geometry, calculus, and physical modeling.

Study Notes

A function $u(x,y)$ is harmonic if $u_{xx}+u_{yy}=0$.
Harmonic functions are smooth and balanced; they often model steady-state systems like temperature or potential.
If $f(z)=u(x,y)+iv(x,y)$ is analytic, then both $u$ and $v$ are harmonic.
The Cauchy-Riemann equations are $u_x=v_y$ and $u_y=-v_x$.
To test harmonicity, compute $u_{xx}$ and $u_{yy}$ and check whether their sum is $0$.
Examples of harmonic functions include $x^2-y^2$, $2xy$, and $e^x\cos y$.
Non-examples include $x^2+y^2$ because $u_{xx}+u_{yy}\neq 0$.
Harmonic conjugates help build analytic functions from harmonic real parts.