Complex Variable Differentiation

Unit 4 — Complex Variable: Differentiation

This unit introduces the fundamentals of complex variables, complex differentiation, analytic functions, Cauchy-Riemann equations, harmonic functions, conformal mappings, and Möbius transformations.

1. Functions of a Complex Variable

A complex variable $z$ can be represented as:

z = x + iy

where:

$x$ = real part of $z$
$y$ = imaginary part of $z$
$i = \sqrt{-1}$

The conjugate of $z$ is:

\bar{z} = x - iy

The real and imaginary parts can be obtained as:

x = \frac{z + \bar{z}}{2}

y = \frac{z - \bar{z}}{2i}

The modulus of $z$ is:

|z| = \sqrt{x^2 + y^2}

The argument of $z$ is:

\arg z = \tan^{-1}\left(\frac{y}{x}\right)

(with the appropriate quadrant considered).

A complex number can also be represented in polar form:

z = r(\cos\theta + i\sin\theta)

where:

r = |z|, \qquad \theta = \arg z

Using Euler’s formula:

e^{i\theta} = \cos\theta + i\sin\theta

the polar form becomes:

z = re^{i\theta}

2. Complex Function

A complex function can be written as:

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

where:

$u(x,y)$ = real part
$v(x,y)$ = imaginary part

Therefore:

\operatorname{Re}[f(z)] = u(x,y), \qquad \operatorname{Im}[f(z)] = v(x,y)

The modulus of $f(z)$ is:

|f(z)| = \sqrt{u^2(x,y) + v^2(x,y)}

3. Limit of a Complex Function

The limit of $f(z)$ as $z$ approaches $z_0$ is written as:

\lim_{z \to z_0} f(z) = L

This means that $f(z)$ approaches $L$ as $z$ approaches $z_0$ from any direction in the complex plane.

If $f(z) = u(x,y) + iv(x,y)$ and $L = L_1 + iL_2$ , then:

\lim_{(x,y)\to(x_0,y_0)} u(x,y) = L_1

\lim_{(x,y)\to(x_0,y_0)} v(x,y) = L_2

4. Continuity

A function $f(z)$ is continuous at $z = z_0$ if:

\lim_{z \to z_0} f(z) = f(z_0)

For $f(z) = u(x,y) + iv(x,y)$ , continuity requires both $u(x,y)$ and to be continuous at .

5. Differentiability

The derivative of a complex function is defined by:

f'(z) = \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z}

provided the limit exists and is independent of the direction in which $\Delta z$ approaches zero.

For $f(z) = u(x,y) + iv(x,y)$ , when the Cauchy-Riemann equations are satisfied:

f'(z) = u_x + iv_x

where

u_x = \frac{\partial u}{\partial x}, \qquad v_x = \frac{\partial v}{\partial x}

6. Analytic Functions

A function $f(z)$ is analytic at a point if it is differentiable at every point in some neighborhood of that point.

A function is analytic in a domain $D$ if it is differentiable at every point of $D$ .

If $f(z) = u(x,y) + iv(x,y)$ and $u,v$ have continuous first-order partial derivatives satisfying the Cauchy-Riemann equations, then is analytic.

7. Cauchy-Riemann Equations

For $f(z) = u(x,y) + iv(x,y)$ , the Cartesian form of the Cauchy-Riemann equations is:

\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}

\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

Equivalently:

u_x = v_y, \qquad u_y = -v_x

8. Derivative Using Cauchy-Riemann Equations

When the Cauchy-Riemann equations are satisfied:

f'(z) = u_x + iv_x

Using the CR equations, this can also be written as:

f'(z) = v_y - iu_y

Thus we have the equivalent forms:

f'(z) = u_x + iv_x = v_y - iu_y

9. Polar Form of Cauchy-Riemann Equations

Let $z = re^{i\theta}$ and $f(z) = u(r,\theta) + iv(r,\theta)$ .

The Cauchy-Riemann equations in polar coordinates are:

\frac{\partial u}{\partial r} = \frac{1}{r}\frac{\partial v}{\partial\theta}

\frac{\partial v}{\partial r} = -\frac{1}{r}\frac{\partial u}{\partial\theta}

Equivalently:

u_r = \frac{1}{r}v_\theta, \qquad v_r = -\frac{1}{r}u_\theta

(for $r \neq 0$ ).

10. Harmonic Functions

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

\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0

u_{xx} + u_{yy} = 0

Similarly, $v(x,y)$ is harmonic if:

v_{xx} + v_{yy} = 0

If $f(z) = u(x,y) + iv(x,y)$ is analytic, then both $u$ and $v$ are harmonic functions.

11. Harmonic Conjugate

If $u(x,y)$ and $v(x,y)$ satisfy the Cauchy-Riemann equations

u_x = v_y, \qquad u_y = -v_x

then $v$ is called the harmonic conjugate of $u$ (and vice versa).

Consequently:

f(z) = u + iv

is an analytic function.

12. Method to Find an Analytic Function

Given the real part $u(x,y)$ :

From Cauchy-Riemann:

v_y = u_x

Integrate with respect to $y$ :

v = \int u_x\, dy + \phi(x)

where $\phi(x)$ is a function of $x$ only.

Use the second Cauchy-Riemann equation $u_y = -v_x$ to determine $\phi(x)$ .

Finally:

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

Given the imaginary part $v(x,y)$ :

Use

u_x = v_y, \qquad u_y = -v_x

and proceed analogously.

13. Milne-Thomson Method

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

f'(z) = u_x - iu_y

After computing the partial derivatives, substitute

y = 0, \quad x = z

to obtain $f'(z)$ . Then integrate:

f(z) = \int f'(z)\, dz + C

where $C$ is a complex constant.

14. Conformal Mapping

A transformation $w = f(z)$ is called conformal at a point if it preserves angles between curves at that point, provided:

f'(z) \neq 0

and $f$ is analytic in a neighborhood of the point.

If two curves intersect at an angle $\theta$ , a conformal mapping preserves both the magnitude and orientation of the angle:

\theta_w = \theta_z

Local scale factor:

|f'(z)|

Local rotation angle:

\arg f'(z)

Hence:

f'(z) = |f'(z)|\, e^{i\arg f'(z)}

15. Möbius Transformation

A Möbius transformation (linear fractional transformation) has the form:

w = \frac{az + b}{cz + d}

where $a,b,c,d$ are complex constants satisfying:

ad - bc \neq 0

16. Derivative of a Möbius Transformation

\frac{dw}{dz} = \frac{ad - bc}{(cz + d)^2}

Since $ad - bc \neq 0$ , the derivative is non-zero wherever $cz + d \neq 0$ .

17. Inverse Möbius Transformation

Starting from $w = \frac{az + b}{cz + d}$ , solving for $z$ yields:

z = \frac{b - dw}{cw - a}

or equivalently:

z = \frac{dw - b}{a - cw}

18. Special Cases of Möbius Transformations

Translation

w = z + b

Rotation and Scaling

w = az

If $a = re^{i\theta}$ , then:

w = re^{i\theta}z

$r$ = scale factor
$\theta$ = rotation angle

Inversion

w = \frac{1}{z}

19. Important Properties of Möbius Transformations

One-to-one (non-degenerate case).
Circles and lines map to circles and lines (generalized circles).
Conformal wherever the derivative exists and is non-zero.
Acts on the extended complex plane $\mathbb{C} \cup \{\infty\}$ .
Pole (when $c \neq 0$ ):

z = -\frac{d}{c}

20. Fixed Points of a Möbius Transformation

A fixed point satisfies $w = z$ :

z = \frac{az + b}{cz + d}

which leads to the quadratic equation:

cz^2 + (d - a)z - b = 0

Key Formulas Summary

\begin{aligned} z & = x + i y \\ \overset{ˉ}{z} & = x - i y \\ ∣ z ∣ & = \sqrt{x^{2} + y^{2}} \\ z & = r e^{i θ} \\ e^{i θ} & = \cos θ + i \sin θ \\ f (z) & = u (x, y) + i v (x, y) \end{aligned}

What You'll Learn

By completing this unit, you will be able to:

Represent complex numbers in Cartesian and polar forms.
Work with complex functions.
Calculate limits of complex functions.
Determine continuity and differentiability.
Understand analytic functions.
Apply the Cauchy-Riemann equations in Cartesian form.
Apply the Cauchy-Riemann equations in polar form.
Identify and construct harmonic functions.
Find harmonic conjugates.
Construct analytic functions from their real or imaginary parts.
Apply the Milne-Thomson method.
Understand conformal mappings.
Calculate local scaling and rotation under conformal mappings.
Work with Möbius transformations.
Find derivatives and inverse transformations of Möbius mappings.
Determine fixed points of Möbius transformations.

Key Takeaway

This unit develops the foundation of complex-variable differentiation. You will learn how to determine analyticity using the Cauchy-Riemann equations, construct analytic functions using harmonic conjugates and the Milne-Thomson method, and understand how conformal and Möbius transformations map regions in the complex plane.