This chapter focuses on the application of complex variables in electronics engineering, a fundamental area for analyzing AC circuits, signal processing, and control systems. Complex variables simplify the representation of sinusoidal signals and system behaviors using real and imaginary parts, enabling more efficient problem-solving. Mastery of functions of complex variables and their derivatives is essential for designing stable and responsive electronic systems in Kenyan industries such as telecommunications, power electronics, and embedded systems.

8.1 Functions of Complex Variable

In electronics engineering, a complex variable is used to represent signals and system responses that have both magnitude and phase components. Functions of complex variables map complex inputs to complex outputs, allowing engineers to analyze frequency responses, impedance, and transfer functions effectively. This section develops the concept of these functions and their derivatives, which are crucial for understanding system behavior in the complex plane.

8.1.1 Functions of a Complex Variable, Concept and Definition

A function of a complex variable \(f(z)\) assigns to each complex number \(z = x + iy\) a complex number \(w = u(x,y) + iv(x,y)\), where \(u\) and \(v\) are real-valued functions of two variables \(x\) and \(y\). The function maps points in the complex plane to other points in the complex plane.

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

Worked Examples

Example 1:
Given \(f(z) = z^2\), find \(f(2 + 3i)\).

Given: \(z = 2 + 3i\)

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

$$ f(2 + 3i) = (2 + 3i)^2 $$

$$ = 2^2 + 2 \times 2 \times 3i + (3i)^2 $$

$$ = 4 + 12i + 9i^2 $$

Since \(i^2 = -1\),

$$ = 4 + 12i - 9 $$

$$ = -5 + 12i $$

Answer: \(-5 + 12i\)

Example 2:
Evaluate \(f(z) = 3z + 4\) at \(z = -1 + 2i\).

Given: \(z = -1 + 2i\)

$$ f(z) = 3z + 4 $$

$$ f(-1 + 2i) = 3(-1 + 2i) + 4 $$

$$ = -3 + 6i + 4 $$

$$ = 1 + 6i $$

Answer: \(1 + 6i\)

Example 3:
Compute \(f(z) = \frac{1}{z}\) for \(z = 1 - i\).

Given: \(z = 1 - i\)

$$ f(z) = \frac{1}{z} $$

To find \(1/(1 - i)\), multiply numerator and denominator by the conjugate:

$$ \frac{1}{1 - i} \times \frac{1 + i}{1 + i} = \frac{1 + i}{(1)^2 - (i)^2} = \frac{1 + i}{1 - (-1)} = \frac{1 + i}{2} $$

$$ = \frac{1}{2} + \frac{i}{2} $$

Answer: \(\frac{1}{2} + \frac{i}{2}\)

8.1.2 Limits and Continuity of Complex Functions

The limit of a complex function \(f(z)\) as \(z\) approaches \(z_0\) exists if the value of \(f(z)\) approaches a fixed complex number \(w_0\) regardless of the path taken in the complex plane. Continuity requires the limit at \(z_0\) to equal \(f(z_0)\).

$$ \lim_{z \to z_0} f(z) = w_0 $$

Worked Examples

Example 1:
Evaluate \(\lim_{z \to 1 + i} (z^2 + 2z)\).

Given: \(z \to 1 + i\)

$$ \lim_{z \to 1 + i} (z^2 + 2z) = (1 + i)^2 + 2(1 + i) $$

Calculate \((1 + i)^2\):

$$ = 1^2 + 2 \times 1 \times i + i^2 = 1 + 2i - 1 = 2i $$

Calculate \(2(1 + i)\):

$$ = 2 + 2i $$

Sum:

$$ 2i + 2 + 2i = 2 + 4i $$

Answer: \(2 + 4i\)

Example 2:
Check if the function \(f(z) = \frac{z^2 - 1}{z - 1}\) is continuous at \(z=1\).

Given: \(z \to 1\)

Simplify:

$$ f(z) = \frac{(z - 1)(z + 1)}{z - 1} = z + 1, \quad z eq 1 $$

Calculate limit:

$$ \lim_{z \to 1} f(z) = \lim_{z \to 1} (z + 1) = 2 $$

Evaluate \(f(1)\):

The function is undefined at \(z=1\) (division by zero).

Since limit exists but \(f(1)\) is undefined, \(f\) is not continuous at \(z=1\).

Answer: Function not continuous at \(z=1\)

Example 3:
Find \(\lim_{z \to 0} \frac{\sin z}{z}\).

Given: \(z \to 0\)

Using the standard limit from calculus:

$$ \lim_{z \to 0} \frac{\sin z}{z} = 1 $$

Answer: 1

8.1.3 Differentiability of Complex Functions

A complex function \(f(z)\) is differentiable at \(z_0\) if the derivative

$$ f'(z_0) = \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h} $$

exists and is the same regardless of the direction from which \(h\) approaches zero in the complex plane. Differentiability in complex analysis is stricter than in real analysis because it must hold for every direction.

Worked Examples

Example 1:
Find the derivative of \(f(z) = z^3\) at \(z = 2 + i\).

Given: \(f(z) = z^3\), \(z = 2 + i\)

Formula for derivative:

$$ f'(z) = 3z^2 $$

Evaluate:

$$ f'(2 + i) = 3(2 + i)^2 $$

Calculate \((2 + i)^2\):

$$ = 4 + 4i + i^2 = 4 + 4i - 1 = 3 + 4i $$

Multiply by 3:

$$ 3(3 + 4i) = 9 + 12i $$

Answer: \(9 + 12i\)

Example 2:
Determine if \(f(z) = |z|^2 = x^2 + y^2\) is differentiable at \(z = 0\).

Given: \(f(z) = x^2 + y^2\), with \(z = x + iy\)

This function is not complex differentiable because it depends on \(x\) and \(y\) separately and does not satisfy the Cauchy-Riemann equations (to be discussed later).

Answer: Not differentiable at \(z = 0\)

Example 3:
Calculate the derivative of \(f(z) = e^z\) at \(z = i\).

Given: \(f(z) = e^z\)

Derivative:

$$ f'(z) = e^z $$

Evaluate at \(z = i\):

$$ f'(i) = e^i = \cos 1 + i \sin 1 $$

Answer: \(\cos 1 + i \sin 1\)

8.1.4 Derivatives of Analytic Functions

An analytic function is a complex function differentiable at every point in an open subset of the complex plane. Its derivative is also analytic in that domain. Analytic functions are central in electronics engineering for stable signal and system analysis.

The derivative of an analytic function \(f(z) = u(x,y) + iv(x,y)\) can be found using:

$$ f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} $$

provided the Cauchy-Riemann equations hold:

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

Worked Examples

Example 1:
Verify if \(f(z) = z^2 = (x + iy)^2\) is analytic and find \(f'(z)\).

Given: \(f(z) = z^2 = (x + iy)^2 = x^2 - y^2 + 2ixy\)

So,

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

Calculate partial derivatives:

$$ \frac{\partial u}{\partial x} = 2x, \quad \frac{\partial u}{\partial y} = -2y $$

$$ \frac{\partial v}{\partial x} = 2y, \quad \frac{\partial v}{\partial y} = 2x $$

Check Cauchy-Riemann:

$$ \frac{\partial u}{\partial x} = 2x \quad \stackrel{?}{=} \quad \frac{\partial v}{\partial y} = 2x $$

True.

$$ \frac{\partial u}{\partial y} = -2y \quad \stackrel{?}{=} \quad -\frac{\partial v}{\partial x} = -2y $$

True.

Calculate derivative:

$$ f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = 2x + 2iy = 2(x + iy) = 2z $$

Answer: \(f'(z) = 2z\)

Example 2:
Check if \(f(z) = \bar{z} = x, iy\) is analytic.

Given:

$$ u = x, \quad v = -y $$

Partial derivatives:

$$ \frac{\partial u}{\partial x} = 1, \quad \frac{\partial u}{\partial y} = 0 $$

$$ \frac{\partial v}{\partial x} = 0, \quad \frac{\partial v}{\partial y} = -1 $$

Cauchy-Riemann:

$$ \frac{\partial u}{\partial x} = 1 \quad \stackrel{?}{=} \quad \frac{\partial v}{\partial y} = -1 $$

False.

Answer: Not analytic

Example 3:
Find the derivative of \(f(z) = e^{z}\) and verify analyticity.

Given \(f(z) = e^{z} = e^{x}(\cos y + i \sin y)\)

So,

$$ u = e^{x} \cos y, \quad v = e^{x} \sin y $$

Partial derivatives:

$$ \frac{\partial u}{\partial x} = e^{x} \cos y, \quad \frac{\partial u}{\partial y} = -e^{x} \sin y $$

$$ \frac{\partial v}{\partial x} = e^{x} \sin y, \quad \frac{\partial v}{\partial y} = e^{x} \cos y $$

Check Cauchy-Riemann:

$$ \frac{\partial u}{\partial x} = e^{x} \cos y = \frac{\partial v}{\partial y} = e^{x} \cos y $$

True.

$$ \frac{\partial u}{\partial y} = -e^{x} \sin y = -\frac{\partial v}{\partial x} = -e^{x} \sin y $$

True.

Derivative:

$$ f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = e^{x} \cos y + i e^{x} \sin y = e^{x}(\cos y + i \sin y) = e^{z} $$

Answer: \(f'(z) = e^{z}\)

Chapter Summary

This chapter focused on functions of a complex variable, introducing the concept and properties that distinguish them from real functions. It then explored the derivatives of analytic functions, emphasizing the conditions under which these derivatives exist and their significance in complex analysis. The chapter further defined analytic, or regular, functions as those that are complex differentiable in a neighborhood of every point in their domain, highlighting their smoothness and structural importance. A critical part of the discussion was the Cauchy-Riemann equations, which provide necessary and sufficient conditions for a complex function to be analytic. Understanding these equations allows for the verification of analyticity by relating partial derivatives of the real and imaginary parts of the function. The chapter combined theoretical explanations with mathematical rigor to build a foundation for further study in complex variables and their applications in engineering contexts.