Complex Variables and Analytic Functions

Introduction and context

Complex-variable theory extends real-variable calculus by introducing the imaginary unit i = √(-1) and numbers of the form z = x + iy, where x and y are real. Although complex numbers may initially appear abstract, they provide an exceptionally efficient language for modelling oscillations, alternating-current circuits, wave propagation, feedback systems, fluid flow and inverse transforms.

In mechatronics, a sinusoidal signal can be represented compactly by a complex exponential. Differentiation of this exponential becomes multiplication by a constant, while integration becomes division. This simplifies the analysis of motors, sensors, filters, vibration systems and electrical networks. Complex analysis also supplies powerful integral theorems that allow difficult real integrals to be evaluated by examining singularities and residues.

This chapter develops the algebra and geometry of complex numbers, differentiability and analyticity, contour integration, Cauchy’s theorems, Taylor and Laurent series, singularities and residues. The emphasis is on conditions of applicability and engineering interpretation.

Learning objectives

After studying this chapter, you should be able to:

1. Represent, manipulate and interpret complex numbers in Cartesian, polar and exponential forms.
2. Determine whether a complex function is differentiable or analytic using the Cauchy–Riemann equations.
3. Evaluate elementary contour integrals and apply Cauchy’s integral theorem and formula.
4. Construct Taylor and Laurent expansions in specified regions.
5. Classify isolated singularities and calculate residues.
6. Use residue methods in oscillation, circuit, field and inverse-transform problems.
7. Recognise the assumptions and limitations of complex-variable methods.

1. Complex numbers and their geometry

A complex number is written as

The real part is Re(z) = x and the imaginary part is Im(z) = y. Two complex numbers are equal if and only if their real and imaginary parts are separately equal.

The complex plane represents z = x + iy by the point (x,y). The modulus and conjugate are

The modulus is the distance from the origin, and the conjugate is reflection in the real axis. Important identities include

The last relationship is the triangle inequality. It is fundamental in estimating errors and proving convergence.

If z ≠ 0, its reciprocal is

Thus, for w = u + iv, division can be performed by multiplying numerator and denominator by the conjugate of the denominator.

Polar and exponential forms

Let r = |z| and let θ = arg(z) be an argument of z. Then

Euler’s formula states

so that z = re^iθ. The argument is not unique: θ + 2kπ, where k ∈ ℤ, represents the same point. A selected principal argument is often denoted Arg(z), commonly restricted to (−π, π].

Multiplication and division become especially simple in polar form:

De Moivre’s formula is

For n > 0, the n roots of z = re^iθ are

The roots are equally spaced on a circle in the complex plane.

Worked example 1: polar multiplication

Let z₁ = 2e^iπ/6 and z₂ = 3e^iπ/3. Then

The modulus is 6 and the argument is π/2. In Cartesian form, the answer is therefore 0 + 6i.

2. Complex functions and limits

A complex function maps a subset of the complex plane into the complex plane. Write

where u and v are real-valued functions. A limit lim_z→z₀ f(z) = L exists only if the same value is obtained as z approaches z₀ along every possible path in the plane. This is more demanding than checking two or three paths, although different path limits can disprove existence.

Continuity at z₀ requires

Polynomials in z, rational functions where the denominator is nonzero, and elementary functions such as e^z, sin z and cos z are continuous on their natural domains.

3. Complex differentiation and analyticity

The complex derivative at z₀ is defined by

provided the limit is independent of the direction of approach. This directional independence is a strong condition. A function is analytic (or holomorphic) in a domain if it has a complex derivative at every point of that domain. It is entire if it is analytic throughout the complex plane.

Cauchy–Riemann equations

Suppose f(z) = u(x,y) + iv(x,y) and the first partial derivatives of u and v are continuous near a point. A necessary condition for differentiability at that point is

These are the Cauchy–Riemann equations. If the partial derivatives are continuous in a neighbourhood and satisfy these equations there, then f is analytic in that neighbourhood. When differentiable,

The continuity qualification matters: satisfying the equations at one point alone is not generally sufficient for analyticity.

Worked example 2: testing analyticity

Consider f(z) = z². Since

we have u = x² − y² and v = 2xy. Therefore

The equations hold everywhere, and the partial derivatives are continuous. Thus z² is entire, with derivative f′(z) = 2z.

By contrast, f(z) = z̄ = x − iy has u = x and v = −y. Here u_x = 1 but v_y = −1, so it is nowhere analytic.

Harmonic functions

If f is analytic, then its real and imaginary parts are harmonic, meaning they satisfy Laplace’s equation:

The pair u and v are called harmonic conjugates when they form an analytic function. This connection is important in two-dimensional electrostatic and fluid-flow models, where potentials satisfy Laplace’s equation.

4. Elementary analytic functions

The exponential function is defined by the convergent series

For z = x + iy,

The complex sine and cosine are defined by

They remain entire, but unlike their real counterparts they are not bounded on the whole complex plane. Complex logarithms are multivalued:

A single-valued principal logarithm requires a branch cut, commonly the non-positive real axis. Branch choices are essential when using powers such as z^α = e^{α Log z}.

5. Contours and contour integration

A smooth parametrised contour is z = z(t), a ≤ t ≤ b, where z(t) is continuously differentiable, except possibly at finitely many points. The contour integral is

For f = u + iv and dz = dx + i dy,

The integral depends on orientation. Reversing a contour changes its sign.

If f has an antiderivative F, then

so the integral is path independent. Every closed-contour integral is then zero.

Cauchy’s integral theorem

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

A simply connected domain has no holes. More general versions permit multiply connected domains provided analyticity holds throughout the region between relevant contours.

Cauchy’s integral formula

If f is analytic on and inside a positively oriented simple closed contour C, and a lies inside C, then

Its derivative form is

for n = 0,1,2,…. The assumptions are crucial: f must be analytic in the enclosed region, while the displayed kernel has its known singularity at a.

Worked example 3: Cauchy’s formula

Let C be the circle |z| = 2, positively oriented, and evaluate

Since i lies inside the circle and e^z is entire, Cauchy’s formula gives

No parametrisation is required.

6. Taylor and Laurent series

If f is analytic in a disk centred at a, it has a Taylor expansion

The radius of convergence extends to the nearest singularity, although the exact domain must be determined from the function.

When a function is analytic in an annulus r₁ < |z − a| < r₂, it has a Laurent expansion

The terms with negative powers form the principal part. A Laurent expansion can differ in different annular regions, so the region must always be stated.

Example: geometric-series expansions

For |z| < 1,

For |z| > 1, rewrite the same function as

The two expansions are both correct, but in different regions.

7. Singularities

An isolated singularity occurs at z = a when a function is analytic in a punctured neighbourhood 0 < |z − a| < R but not defined or not analytic at a.

• A removable singularity has a finite limit as z → a. Its Laurent principal part is absent, and defining the function by that limit removes the singularity.
• A pole of order m has a finite principal part ending in (z − a)^−m, with a nonzero coefficient of that term.
• An essential singularity has infinitely many negative-power terms. Its local behaviour can be highly irregular.

A zero of order m at a means f(z) = (z − a)^m g(z), where g is analytic and g(a) ≠ 0. Zeros and poles are useful in transfer-function analysis because they determine local magnitude and phase behaviour.

8. Residues and the residue theorem

The residue of f at a is the coefficient of (z − a)⁻¹ in its Laurent expansion:

For a simple pole,

If f(z) = g(z)/h(z), where g and h are analytic, g(a) ≠ 0, h(a) = 0 and h′(a) ≠ 0, then a is a simple pole and

For a pole of order m,

The residue theorem states that if f is analytic except at finitely many isolated singularities a₁,…,a_N inside a positively oriented closed contour C, then

Worked example 4: residues

Evaluate the integral around a contour enclosing both poles:

The poles are z = i and z = −i. For a simple pole at a, with h(z) = z² + 1,

Thus the residues are 1/(2i) and −1/(2i), whose sum is zero. Therefore

This result assumes the contour encloses both poles and no other singularities. A contour enclosing only i would instead give 2πi · 1/(2i) = π.

9. Engineering applications

Phasors and oscillations

A real sinusoid can be represented as the real part of a complex exponential:

where A is amplitude, ω is angular frequency in radians per second, t is time in seconds and φ is phase in radians. Differentiation gives

Consequently, velocity and acceleration of harmonic motion correspond to multiplication by iω and −ω², respectively. The physical signal is obtained by taking the real part at the end; the complex quantity is a computational representation.

AC circuits

In sinusoidal steady state, impedance is complex. For angular frequency ω:

where R is resistance in ohms, L inductance in henries and C capacitance in farads. These expressions assume linear, time-invariant components and steady sinusoidal excitation. Kirchhoff’s laws can then be applied using complex voltage and current phasors.

Fields and potential flow

For two-dimensional ideal potential flow or electrostatics, an analytic complex potential can combine a scalar potential and a stream function. The Cauchy–Riemann equations ensure that the corresponding field components satisfy the required orthogonality and, under suitable source-free assumptions, Laplace’s equation. Boundary geometry can sometimes be transformed by a conformal map, which preserves angles locally.

Inverse transforms and modal response

Laplace and Fourier inversion often produces contour integrals. Poles of a transfer function identify exponential modes. A pole at s = σ + iω corresponds to a term proportional to e^σtcos(ωt) or e^σtsin(ωt). Thus negative real parts indicate decay, zero real parts sustained oscillation, and positive real parts growth in the ideal model. Physical stability conclusions require attention to initial conditions, nonlinearities, sampling and model validity.

10. Illustrative mechatronics case: resonant vibration sensor

Background

Consider a linear mass–spring–damper sensor subject to harmonic force F(t) = F₀ cos(ωt). Its model is

where m is mass in kilograms, c damping coefficient in newton-seconds per metre, k stiffness in newtons per metre and x displacement in metres.

Analysis and approach

Represent the force and steady-state displacement by phasors. With x(t) = Re[Xe^iωt], substitution gives

Therefore the mechanical impedance is

and

The displacement amplitude is

This formula assumes a linear model, harmonic steady state and constant parameters. Near the natural frequency ω_n = √(k/m), the amplitude may become large if damping is small. The phase is −arg Z(ω).

Outcome and learning points

A controller or signal-conditioning circuit can use the predicted amplitude and phase to identify resonance, estimate damping or avoid damaging excitation. Complex notation does not change the physical displacement; it organizes amplitude and phase calculations. Model uncertainty, sensor saturation and nonlinear stiffness must be checked before applying the prediction to hardware.

11. Limitations, good practice and contemporary tools

Complex analysis is exact for the mathematical model, but engineering models may be approximate. Common limitations include nonlinearity, discontinuous material properties, noise, finite bandwidth, parameter uncertainty and inappropriate boundary conditions. A pole calculation based on a linearised model may not predict large-signal behaviour.

Good practice includes:

• state the contour orientation and region of analyticity;
• identify every singularity before applying a theorem;
• distinguish a principal argument from a multivalued argument;
• verify convergence regions for series;
• retain units in engineering substitutions;
• use symbolic software to check, not replace, reasoning;
• validate complex frequency-domain results against time-domain simulation or measurement.

Modern computer algebra and numerical packages calculate residues, evaluate contour integrals and generate Bode or Nyquist plots. AI-assisted tools can help discover algebraic errors or suggest transformations, but their outputs require independent verification of domains, branch cuts, poles and physical assumptions. In safety-critical mechatronic systems, traceable calculations and experimentally validated models remain essential.

Chapter Summary

Complex numbers combine two real dimensions into a form especially suited to rotation and oscillation. Cartesian, polar and exponential representations support different tasks: Cartesian form is convenient for addition, while polar form simplifies multiplication, division and powers. A complex function is written in terms of real and imaginary components, and complex differentiability requires direction-independent limits. Under appropriate continuity assumptions, the Cauchy–Riemann equations characterise analyticity.

Analytic functions possess antiderivatives locally and obey Cauchy’s integral theorem and formula. Taylor series describe analytic behaviour near ordinary points; Laurent series also represent isolated singularities in annular regions. Singularities are classified as removable, poles or essential, and residues provide the coefficient needed by the residue theorem. These results transform many contour integrals into finite algebraic sums.

In mechatronics, complex exponentials simplify harmonic motion, phasor circuit analysis, field models and inverse-transform calculations. The location of poles gives important information about transient modes and stability, but conclusions must remain consistent with the assumptions and limits of the underlying physical model.

Key Terms

Complex number: A number of the form x + iy, with real x,y.

Complex plane: Geometric plane representing real and imaginary parts.

Modulus: Magnitude |z| of a complex number.

Conjugate: Number z̄ obtained by reversing the sign of the imaginary part.

Argument: An angle locating a complex number in polar form.

Analytic function: A function complex-differentiable throughout a domain.

Entire function: A function analytic throughout the complex plane.

Cauchy–Riemann equations: Conditions relating the partial derivatives of real and imaginary components.

Harmonic function: A function satisfying Laplace’s equation.

Contour: An oriented curve in the complex plane.

Contour integral: Integral of a complex function along a contour.

Simply connected domain: A domain without holes.

Cauchy integral formula: Formula expressing an analytic function or derivative through a contour integral.

Taylor series: Power series containing non-negative powers about an ordinary point.

Laurent series: Power series allowing both positive and negative powers.

Isolated singularity: A singular point surrounded by a punctured analytic neighbourhood.

Pole: An isolated singularity with a finite nonzero principal part.

Residue: Coefficient of the inverse first-power term in a Laurent expansion.

Phasor: Complex representation of sinusoidal amplitude and phase.

Impedance: Complex opposition to sinusoidal current, measured in ohms.

Review Questions

Short Answer Questions

1. Define the modulus and conjugate of z = x + iy.
2. State the Cauchy–Riemann equations.
3. What is the difference between a Taylor series and a Laurent series?
4. State Cauchy’s integral formula for f(a).
5. What physical information is associated with the real part of a pole in a linear dynamic model?

Long Answer Questions

1. Derive the polar multiplication and division rules from Euler’s formula.
2. Explain complex differentiability and analyticity, including the role of the Cauchy–Riemann equations.
3. State and discuss the hypotheses of Cauchy’s integral theorem and Cauchy’s integral formula.
4. Explain the classification of isolated singularities and show how residues arise from Laurent series.
5. Describe how phasor notation is used to analyse an AC circuit containing resistance, inductance and capacitance.

Analytical / Application Questions

1. Determine whether f(z) = x² − y² + i2xy is analytic, and find its derivative.
2. Evaluate ∮_C dz/(z−2) when C is a positively oriented circle enclosing 2, explaining which theorem applies.
3. For a mass–spring–damper system, explain how the complex impedance predicts resonance and how damping changes the response amplitude and phase.

Multiple Choice Questions with Answers

Question 1

For z = 3 − 4i, what is |z|?

A. 1

B. 5

C. 7

D. 25

Correct Answer: B. 5

Question 2

Which condition is required by the Cauchy–Riemann equations?

A. u_x = v_y and u_y = −v_x

B. u_x = −v_y and u_y = v_x

C. u_x = u_y and v_x = v_y

D. u_x = v_x and u_y = v_y

Correct Answer: A. u_x = v_y and u_y = −v_x

Question 3

What is the polar form of −1 using the principal argument?

A. e^iπ/2

B. e^iπ

C. 2e^iπ

D. e^−iπ/2

Correct Answer: B. e^iπ

Question 4

If f is analytic inside and on a closed contour, what is its integral around that contour?

A. Always 2πi

B. The maximum value of f

C. Zero

D. The residue at the centre

Correct Answer: C. Zero

Question 5

What is the residue of 1/(z−a) at z=a?

A. 0

B. 1

C. a

D. 2πi

Correct Answer: B. 1

Question 6

The Laurent expansion of 1/(1−z) as a series in negative powers applies when:

A. |z| < 1

B. |z| = 1 only

C. |z| > 1

D. For all complex z

Correct Answer: C. |z| > 1

Question 7

For a simple pole of g(z)/h(z) at a, the residue is:

A. g′(a)/h(a)

B. g(a)/h′(a)

C. h′(a)/g(a)

D. g(a)h′(a)

Correct Answer: B. g(a)/h′(a)

Question 8

The impedance of an ideal inductor at angular frequency ω is:

A. R

B. 1/(iωL)

C. iωL

D. −iωC

Correct Answer: C. iωL

Question 9

A pole at s = σ + iω contributes an oscillatory mode whose envelope is proportional to:

A. e^σt

B. e^ωt

C. e^it

D. e^−σt only

Correct Answer: A. e^σt

Question 10

Which singularity has an infinite principal part in its Laurent expansion?

A. Removable singularity

B. Simple pole

C. Essential singularity

D. Ordinary analytic point

Correct Answer: C. Essential singularity