Complex Numbers and Complex Variables

Introduction

Complex numbers provide the natural language for sinusoidal steady-state circuits, electromagnetic fields, communication systems and feedback control. A complex number simultaneously represents magnitude and phase, while a complex variable extends differentiation and integration to two-dimensional domains. This extension produces powerful results—such as Cauchy’s integral formula and the residue theorem—that have no direct counterpart in elementary real-variable calculus.

In electrical engineering, the imaginary unit is usually written j = √{-1} because i commonly denotes current. Mathematics texts often use i; both symbols have the same meaning. This chapter uses j in circuit and system applications and i elsewhere.

Learning Objectives

After studying this chapter, students should be able to:

• convert complex numbers among rectangular, polar and exponential forms;
• apply De Moivre’s formula and calculate complex roots;
• interpret equations and inequalities as loci in the complex plane;
• use phasors to analyze AC circuits;
• test complex functions for analyticity using the Cauchy–Riemann equations;
• evaluate contour integrals using parametrization, Cauchy’s formula and residues;
• apply conformal mappings to field and boundary-value problems;
• interpret poles, zeros, frequency response and stability in system analysis; and
• validate symbolic or numerical complex-variable calculations.

1. Complex Numbers and the Argand Plane

A complex number has the rectangular form

where x = Re(z) is the real part, y = Im(z) is the imaginary part, and i² = -1. The ordered pair (x,y) represents a point in the complex, or Argand, plane.

Two complex numbers are equal only when their corresponding real and imaginary parts are equal. Addition and subtraction are therefore componentwise.

The complex conjugate and modulus are

Consequently,

For z ≠ 0, division may be performed by multiplying numerator and denominator by the conjugate:

1.1 Polar and exponential forms

If z ≠ 0, let r = |z| > 0 and let θ be an argument of z. Then

The angle is multivalued: if θ is an argument, then so is θ+2kπ for every integer k. A selected interval, commonly (-π,π], defines the principal argument Arg(z). The argument of zero is undefined.

For nonzero complex numbers, multiplication and division become

Thus magnitudes multiply or divide, while phases add or subtract.

Worked Example 1: Form conversion

Convert z=-3+3√3i to polar and exponential forms.

The point is in quadrant II, and its reference angle is π/3. Hence Arg(z)=2π/3, giving

A general argument is 2π/3+2kπ, where k∈ℤ.

2. Powers, Roots and Loci

2.1 De Moivre’s formula

For an integer n,

For n≥0, this includes the zero power in the usual way. For negative integers, the base must be nonzero; equivalently, zⁿ=rⁿe^inθ for negative n requires z≠0 and r>0.

2.2 The nth roots

For w=Re^iφ≠0, the solutions of zⁿ=w, where n is a positive integer, are

These roots have equal magnitude and are separated by 2π/n. For w=0, the only root is z=0.

Worked Example 2: Cube roots

The cube roots of 8 are

Therefore,

Each value cubes to 8.

2.3 Complex-plane loci

A locus is the set of points satisfying a complex equation or inequality.

Worked Example 3: An Apollonius circle

Find the locus |z-1|=2|z+1|. Put z=x+iy and square both sides:

Expansion and completion of the square give

The locus is a circle centered at (-5/3,0) with radius 4/3. Squaring is valid because both sides are nonnegative distances.

3. Phasors and AC Circuit Analysis

For sinusoidal steady-state analysis at a single angular frequency ω in rad/s, a real voltage

may be represented by the RMS phasor

The time factor e^jωt is suppressed. Phasor analysis applies to linear time-invariant circuits after transients have decayed; it does not directly combine signals of different frequencies.

The impedances are

Here R is in ohms, L in henries, C in farads and ω in rad/s, so every impedance has units of ohms. Complex Ohm’s law is

With RMS phasors, complex power is

where S is in volt-amperes, real power P is in watts and reactive power Q is in vars. The power factor is P/|S|=cos φ, where φ is the impedance phase for a passive load.

Worked Example 4: Series RL circuit

A 100 V RMS, 50 Hz source supplies a series combination of R=20 Ω and L=0.10 H. Since ω=2πf=100π rad/s,

Thus

The current is

The current lags the voltage, as expected for an inductive load. The real power is approximately P=I²R=(2.685)²(20)=144.2 W.

4. Functions of a Complex Variable

A complex function can be written

where u and v are real-valued functions. A complex derivative at z₀ is defined by

Unlike a real derivative, the limit must be identical for every direction by which Δz approaches zero.

4.1 Cauchy–Riemann equations

If f is differentiable at a point and the first partial derivatives exist there, then

These are the Cauchy–Riemann equations. If the first partial derivatives are continuous in a neighborhood and satisfy these equations throughout that neighborhood, then f is analytic there. Merely satisfying them at one isolated point does not generally establish analyticity in a neighborhood.

When analytic,

An analytic function is also called holomorphic. A function analytic over the entire complex plane is entire.

4.2 Harmonic functions and harmonic conjugates

If f=u+iv is analytic and has continuous second derivatives, both components satisfy Laplace’s equation:

Such functions are harmonic. In two-dimensional electrostatics, steady heat flow and ideal fluid flow, harmonic functions represent potentials satisfying source-free field equations.

Worked Example 5: Constructing an analytic function

Let u=x²-y². Because

it is harmonic. From the Cauchy–Riemann equations,

Integrating v_y=2x with respect to y gives v=2xy+g(x). Then v_x=2y+g′(x)=2y, so g′(x)=0. Therefore

where C is a real constant.

5. Contours and Complex Integration

A contour C is a piecewise smooth parametrized curve z=z(t), a≤t≤b. Its orientation is the direction of increasing t. For continuous f,

Reversing orientation changes the sign of the integral.

Figure: At the rightmost point of a circle, upward motion is counterclockwise and downward motion is clockwise.

5.1 Cauchy’s theorem and integral formula

If f is analytic throughout a simply connected domain D, then for every closed contour C lying in D,

More generally, holes and singularities must be handled through appropriate domain conditions.

If f is analytic on and inside a positively oriented simple closed contour C, and a lies inside C, then Cauchy’s integral formula states

Its derivative form is

The hypotheses concerning analyticity, enclosure and positive orientation are essential.

Worked Example 6: Cauchy integral formula

Let C be the circle |z-2|=1, traversed counterclockwise. Then

The singularity z=2 is inside the contour, and the constant numerator is analytic everywhere. A clockwise traversal would give the negative value.

5.2 Estimation lemma

If |f(z)|≤M on a contour of length L, then

The ML inequality provides a bound, not generally the exact integral.

6. Laurent Series, Singularities and Residues

Around z=a, a function analytic in an annulus may have a Laurent expansion

The coefficient c_-1 is the residue:

Important isolated singularities are:

• Removable singularity: the principal part is zero and a finite analytic extension exists.
• Pole of order m: finitely many negative powers occur, ending with (z-a)^-m.
• Essential singularity: infinitely many negative powers occur.

For a simple pole,

For a pole of order m,

If f(z)=p(z)/q(z), q(a)=0, q′(a)≠0 and p and q are analytic near a, then the pole is simple and

6.1 Residue theorem

If f is analytic on and inside a positively oriented simple closed contour except for finitely many isolated singularities a_k inside it, then

Worked Example 7: Selecting enclosed poles

Evaluate

for counterclockwise orientation. The poles are at 0 and 2, but only 0 lies inside |z|=1. Its residue is

Therefore,

Including the exterior pole at 2 would be an incorrect application of the theorem.

7. Conformal Mapping and Field Applications

An analytic mapping w=f(z) is locally conformal—angle preserving in magnitude and orientation—at points where f′(z)≠0. Conformality is local; global one-to-one behavior requires additional conditions.

Common mappings include:

The bilinear map

maps the right half-plane Re(z)>0 onto the unit disk |w|<1. Indeed,

Conformal mapping is useful in two-dimensional electrostatics because an analytic complex potential F(z)=φ(x,y)+iψ(x,y) has a harmonic potential φ and stream function ψ. Equipotential and field lines intersect orthogonally wherever the derivative is nonzero. A difficult electrode geometry can sometimes be mapped to a strip or disk, solved there, and mapped back. Assumptions usually include a two-dimensional, linear, homogeneous medium and regions free of distributed charge.

8. Complex Variables in Communication and Control Systems

8.1 Transfer functions, poles and zeros

For a continuous-time linear time-invariant system with zero initial conditions, the transfer function is

Here s has units of s⁻¹, and N and D are usually polynomials arranged so that H has the required physical units. Zeros are roots of N(s), while poles are roots of D(s). Poles determine natural modes e^s_kt. A rational causal system is asymptotically stable when all uncancelled poles lie strictly in the open left half-plane.

Pole-zero cancellation should not be accepted uncritically. An unstable pole that appears cancelled algebraically may remain internally through modeling error or hidden state dynamics.

The frequency response is obtained, when meaningful, by evaluating

Its modulus gives gain and its argument gives phase shift. For stable systems, frequency response links complex-variable analysis directly to filters, communication channels and feedback loops.

8.2 Argument principle and Nyquist reasoning

If F is meromorphic inside a closed contour and has no zeros or poles on it, then

where Z and P count zeros and poles inside the positively oriented contour, including multiplicity. Equivalently, the net change in argument of F(s) is 2π(Z-P).

In feedback analysis, F(s)=1+L(s), where L is the loop transfer function. Zeros of F are closed-loop poles. Nyquist analysis maps a contour through L(s) and studies winding around -1. The sign of an encirclement count depends on the chosen contour and orientation convention, so a convention must be stated rather than inferred from a sketch.

8.3 Discrete-time systems

For a discrete-time system, the transfer function uses the variable z. A causal rational system is asymptotically stable if every uncancelled pole lies inside the unit circle |z|<1. Frequency response is evaluated on z=e^jΩ, where Ω is digital angular frequency in radians per sample. This unit-circle interpretation is central to digital filters and sampled communication receivers.

9. Illustrative Case: Feedback Amplifier Analysis

Background

This fictional case considers a sensor amplifier whose plant model is

with unity negative feedback and proportional controller K=1. The model is assumed linear, time invariant and valid over the operating range.

Problem

Determine the closed-loop poles, stability, damping characteristics, steady-state gain and response at ω=2 rad/s.

Analysis and solution

The loop transfer function is L(s)=G(s). The closed-loop transfer function is

The poles satisfy s²+5s+14=0, so

Thus

Both poles have negative real parts, so the closed loop is asymptotically stable. Comparison with s²+2ζω_n s+ω_n² gives

The DC gain is

At s=j2,

Therefore,

A sinusoidal input at 2 rad/s is therefore reduced to approximately 70.7% of its amplitude and shifted by −45°.

Outcome

Complex arithmetic identifies not only stability but also transient damping and sinusoidal response. The result is valid only to the extent that the transfer-function model represents the physical amplifier.

Learning points

• Pole locations connect complex geometry to transient behavior.
• Evaluation on the imaginary axis gives frequency response.
• Feedback modifies both poles and steady-state gain.
• Stability conclusions require all relevant internal dynamics and parameter uncertainty to be considered.

10. Computational Practice, Limitations and Professional Responsibility

Engineering software can calculate roots, residues, contour plots and frequency responses, but numerical results require verification. Branch choices for logarithms, roots and inverse trigonometric functions can produce discontinuities. Nearly cancelling poles and zeros may be ill-conditioned, while numerical contour integration can fail near singularities.

Recommended practice includes:

1. state the branch and argument convention;
2. identify singularities before integrating;
3. check contour orientation and enclosed poles;
4. verify units in circuit and system models;
5. compare symbolic results with numerical sampling;
6. test sensitivity to uncertain component values; and
7. preserve model assumptions and software settings for reproducibility.

AI-assisted algebra may suggest transformations or code, but it does not establish theorem conditions or model validity. In safety-critical communication, medical or control equipment, unverified pole-zero cancellation, incorrect contour orientation or unstable controller deployment may cause harm. Professional responsibility therefore requires independent validation, traceable calculations and compliance with relevant engineering safety and electromagnetic-compatibility standards.

Contemporary applications include complex baseband signal processing, impedance spectroscopy, electromagnetic simulation, robust control and conformal-mapping-based computational grids. Numerical methods extend classical theory to irregular geometries, but they complement rather than replace analytic reasoning.

Chapter Summary

Complex numbers can be expressed in rectangular, polar and exponential forms. Polar form simplifies multiplication, division, powers and roots, while complex loci translate algebraic conditions into geometric objects. Phasors apply these ideas to sinusoidal steady-state circuit analysis.

Complex differentiability requires direction-independent limits and is characterized, under suitable regularity conditions, by the Cauchy–Riemann equations. Analytic functions have harmonic real and imaginary parts. Contour integration, Cauchy’s integral formula and the residue theorem evaluate integrals through analyticity and singularity structure, with contour orientation and enclosure conditions being essential.

Conformal mappings transform two-dimensional field problems while preserving local angles where the derivative is nonzero. In communication and control systems, poles, zeros, frequency response, the argument principle and unit-circle geometry connect complex analysis to stability and system performance. Reliable engineering use requires careful attention to branches, assumptions, units, uncertainty and computational validation.

Key Terms

Analytic function: A function complex differentiable throughout an open set.

Argument: An angle specifying the direction of a nonzero complex number.

Bilinear transformation: A map (az+b)/(cz+d) with ad-bc≠0.

Cauchy–Riemann equations: Partial differential conditions relating the real and imaginary parts of an analytic function.

Complex conjugate: The number obtained by reversing the sign of the imaginary part.

Conformal mapping: A locally angle-preserving analytic mapping where its derivative is nonzero.

Contour: A directed, piecewise smooth curve in the complex plane.

Entire function: A function analytic on the whole complex plane.

Harmonic function: A twice differentiable function satisfying Laplace’s equation.

Impedance: Complex ratio of phasor voltage to phasor current.

Laurent series: A series containing positive and negative integer powers about a point.

Modulus: The nonnegative magnitude |z| of a complex number.

Phasor: A complex representation of a sinusoid’s RMS magnitude and phase.

Pole: An isolated singularity where a function diverges with finite algebraic order.

Residue: The coefficient of (z-a)^-1 in a Laurent expansion.

Singularity: A point at which a function is not analytic.

Transfer function: The zero-initial-condition ratio of output transform to input transform.

Zero: A point at which a function or transfer-function numerator vanishes.

Review Questions

Short Answer Questions

1. Distinguish between an argument and the principal argument of a complex number.
2. State the condition required when applying De Moivre’s formula with a negative integer exponent.
3. Write the Cauchy–Riemann equations in Cartesian coordinates.
4. Define the residue of a function at an isolated singularity.
5. Where must the poles of a causal rational discrete-time system lie for asymptotic stability?

Long Answer Questions

1. Derive the formula for the n distinct roots of a nonzero complex number and explain their geometry.
2. Explain phasor analysis and derive the impedances of resistors, inductors and capacitors from their time-domain relationships.
3. Discuss complex differentiability, analyticity, harmonic functions and harmonic conjugates.
4. State and compare Cauchy’s theorem, Cauchy’s integral formula and the residue theorem, including their conditions.
5. Explain how conformal mapping can simplify a two-dimensional electrostatic boundary-value problem.

Analytical / Application Questions

1. A 120 V RMS source at 60 Hz supplies a series circuit with R=30 Ω and C=100 μF. Determine the impedance, current magnitude, phase and real power.
2. Evaluate ∮_C (z+2)/[(z-1)(z+3)]dz when C is the counterclockwise circle |z|=2. Clearly identify the enclosed singularities.
3. For a unity-feedback system with G(s)=K/[s(s+3)], derive the closed-loop poles and determine the range of positive K for asymptotic stability. Interpret the pole motion as K increases.

Multiple Choice Questions with Answers

Question 1

What is the principal argument of z=-1+i√3 on the interval (-π,π]?

A. π/3

B. 2π/3

C. 4π/3

D. -2π/3

Correct Answer: B. 2π/3

Question 2

Which condition is required when De Moivre’s formula is used with a negative integer exponent?

A. The complex number must be real.

B. The argument must be positive.

C. The complex number must be nonzero.

D. The modulus must equal one.

Correct Answer: C. The complex number must be nonzero.

Question 3

The locus |z-(2+i)|=3 is which geometric object?

A. A circle centered at 2+i with radius 3

B. A circle centered at -2-i with radius 3

C. A disk centered at 2+i

D. A line through 2+i

Correct Answer: A. A circle centered at 2+i with radius 3

Question 4

At angular frequency ω, what is the impedance of a capacitor C?

A. jωC

B. -jωC

C. j/(ωC)

D. 1/(jωC)

Correct Answer: D. 1/(jωC)

Question 5

For u=x²-y² and v=2xy, which function is represented?

A. z²

B. z̄²

C. |z|²

D. 1/z

Correct Answer: A. z²

Question 6

For a counterclockwise contour enclosing z=2, what is ∮(1+i)/(z-2)dz?

A. 2π+2πi

B. -2π-2πi

C. -2π+2πi

D. 2π-2πi

Correct Answer: C. -2π+2πi

Question 7

Which singularity lies inside the contour |z|=1 for f(z)=(z+1)/[z(z-2)]?

A. Only z=0

B. Only z=2

C. Both z=0 and z=2

D. Neither singularity

Correct Answer: A. Only z=0

Question 8

At which points is an analytic mapping w=f(z) locally conformal?

A. Where f(z)=0

B. Where f′(z)≠0

C. Only where Im(z)=0

D. Only where |f(z)|=1

Correct Answer: B. Where f′(z)≠0

Question 9

For a causal rational continuous-time system, which pole set gives asymptotic stability?

A. All uncancelled poles have negative real parts.

B. All poles lie on the imaginary axis.

C. At least one pole has a positive real part.

D. All poles have magnitudes greater than one.

Correct Answer: A. All uncancelled poles have negative real parts.

Question 10

For the case-study system T(s)=10/(s²+5s+14), what is its DC gain?

A. 5/7

B. 1

C. 7/5

D. 2/7

Correct Answer: A. 5/7