Engineering Math Series 22: Complex Numbers and Analytic Functions

Complex numbers may feel like "strange numbers that do not exist on the real line" when first learned, but in engineering they are actually a very natural tool. This is because they can express oscillation, rotation, and frequency response in the most concise way.

Two Faces of Complex Numbers

A complex number can be written as

$z=x+iy$

or as

$z=re^{i\theta}$

The first is the coordinate representation, and the second is the magnitude-and-angle representation. In engineering, the second is especially important because multiplication becomes "multiply magnitudes, add angles."

Why Are They Useful in Engineering

Rather than handling sines and cosines separately every time for oscillation,

$e^{i\omega t}$

greatly simplifies calculations. AC circuit phasors, Fourier transforms, and Laplace transforms are all connected to this perspective.

Complex Functions

A function defined on the complex variable $z$ can be split into real and imaginary parts as

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

At the introductory level, the important question is "when is differentiation of complex functions well-defined."

Analytic Functions and Cauchy-Riemann

For a complex function to be analytic is a much stronger condition than merely being differentiable. Approaching from any direction must yield the same derivative.

The conditions that appear here are the Cauchy-Riemann equations.

$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

For beginners, understanding this as "the real part and imaginary part are not arbitrary but are very precisely linked" is a good starting point.

Worked Example

Consider the function

$f(z)=z^2$

Substituting

$z=x+iy$

gives

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

so

$u(x,y)=x^2-y^2, \quad v(x,y)=2xy$

Taking partial derivatives,

$u_x=2x, \quad v_y=2x$

$u_y=-2y, \quad -v_x=-2y$

so the Cauchy-Riemann conditions are satisfied. Therefore $z^2$ is an analytic function.

Engineering Applications

AC Circuits

Complex numbers are very convenient for simultaneously capturing the magnitude and phase of voltage and current.

Control and Signal Processing

The form $s=\sigma+i\omega$ expresses damping and oscillation at once.

Fluid and Electromagnetics

In 2D potential problems, analytic functions connect to actual physical fields.

Common Mistakes

Viewing complex numbers only as "imaginary number arithmetic"

In practice, they are a much more natural representation for handling rotation and oscillation.

Only memorizing Cauchy-Riemann

Analytic functions mean the real and imaginary parts are strongly coupled, and this leads to powerful integral theorems.

Ignoring the polar representation

For multiplication, division, and phase analysis, the polar form is much more intuitive.

One-Line Summary

Complex numbers are a natural language for handling oscillation and rotation, and analytic functions are complex functions with very strong structure.

Next Post Preview

In the next post, we will organize the power of complex analysis: complex integration and the residue idea at the introductory level.

References

• Erwin Kreyszig, Advanced Engineering Mathematics, 10th Edition
• James Ward Brown, Ruel V. Churchill, Complex Variables and Applications
• Elias M. Stein, Rami Shakarchi, Complex Analysis