Mastering Complex Calculus: From Derivatives to Residues

Name: Mastering Complex Calculus: From Derivatives to Residues
Rating: 4.7 (71 reviews)

Complex Calculus: derivatives of complex variables, contour integration, Laurent series, Fourier series, and residues

Created byEmanuele Pesaresi

Last updated 5/2026

English

What you'll learn

How to derive the most important theorems and concepts related to Complex Calculus
Cauchy's integral theorem
Cauchy's integral formula
Laurent Series
How to derive the Fourier Series from the Laurent Series
How to derive the Taylor Series from the Laurent Series
Residues
Contour integration

Course content

7 sections • 45 lectures • 11h 7m total length

functions of a complex variable part 18:55
functions of a complex variable part 220:20
Explore complex functions, including conjugates, polynomials, and ratios, and the complex exponential, deriving e^Z = e^X(cos Y + i sin Y) and showing F(Y) is constant.
Concept of derivative in complex calculus19:53
Explore complex calculus by extending derivatives to complex plane, define F′(Z) via limits as W approaches Z, and show differentiability requires path independence, yielding U_x=V_y and U_y=-V_x with harmonic equations.

Concept of Residue7:03
Residue Theorem10:39
Learn how to apply the residue theorem to evaluate contour integrals by summing residues at poles, including simple poles, and extend to multiple discontinuities with contour deformations.
Calculation of residues and coefficients of the Laurent series22:56
Evaluation of a real integral using complex integration (exercise 1)25:19
Contour integration to evaluate a real integral (exercise 2)30:54
Contour integration to evaluate a real integral (exercise 3)25:14
Another integral evaluated using the results of Complex Calculus (exercise 4)7:36
Contour integration to evaluate a complex integral (exercise 5)27:02
Apply contour integration along the imaginary axis to evaluate the principal value of e^(lambda z)/(z(z-2)), handling poles at 0 and 2 with lambda-dependent contour choices and residues.
Another contour integration of a real integral - Exercise 615:36
Fresnel integral over the real line (formally derived with the residue theorem)14:19
This lecture derives the Fresnel integral over the real line using complex calculus and residue theorem, linking cos(t^2) and sin(t^2) to a Gaussian integral via a pi/4 contour and substitution.
Hilbert transform and its geometric meaning15:42
Solution to the diffusion equation using complex calculus and Laplace transform13:26
Representation of the Dirac Delta32:11
Abel-Plana formula in complex Calculus17:34
Convolution of sinc functions using complex calculus27:06
Explore how complex calculus proves that the convolution of two sinc functions is proportional to a sinc, using trigonometric identities and residue techniques, with Fourier transform perspectives.

Complex calculus to evaluate divergent series39:09
Introduction to the analytic continuation of the Riemann zeta function2:27
Train of impulses expanded in a Fourier series8:42
Poisson summation formula4:48
Application of Poisson summation formula6:55
Another representation of the Riemann zeta function5:15
Functional equation of the Riemann zeta function13:53
Extend the Riemann zeta function beyond its initial domain using gamma representations and Poisson summation, and derive a functional equation linking zeta(s) and zeta(1-s).
Evaluation of the Riemann zeta function at s=-37:17
Derives the Riemann zeta functional equation for s with real part less than one and evaluates zeta(-3) using gamma relations and zeta(4)=pi^4/90, showing zeta(-3)=1/120.

Contour integral related to the Euler reflection formula16:09
Derivation of Euler reflection formula5:44
Lagrange duplication formula6:48
Relation between the Beta function and the Gamma function5:49
Another form of the Riemann functional equation8:53
Apply the Riemann functional equation and the Lagrange duplication formula to recast the zeta functional equation, then use Euler's reflection formula to derive an alternative form for zeta(s).
Derivative of the Riemann Zeta function at s=016:53
Derivative of the Gamma function near the point s=15:55
Derive gamma function behavior near epsilon zero by differentiating Gamma(epsilon) and using integration by parts. Identify the Euler–Mascheroni constant gamma from the limit with ln x and e^{-x}, with Re(epsilon)>0.
Representations of the Euler Mascheroni constant7:13
Explore representations of the Euler-Mascheroni constant, linking its integral and limit forms to the discrete harmonic series and the area under 1/x, and note its relevance to physics.

Requirements

Single Variable Calculus (especially: derivatives, integrals)
Multivariable Calculus (especially: Stokes' theorem, line integrals)

Description

This course provides students with a foundation in complex functions, derivatives of complex variables, contour integration, Laurent series, Fourier series, and residues. In this course, you will learn the key concepts of Complex Calculus, and the process of reasoning by using mathematics, rather than rote memorization of formulas and exercises. Here's what you need to know about this course:

Introduction to Complex Functions: The course begins by focusing on the concept of complex functions.

Derivatives of Complex Variables: Next, the concept of derivative is extended to functions of a complex variable.

Contour Integration: You will learn about contour integration, and the following theorems will be derived: Cauchy's integral theorem and Cauchy's integral formula.

Laurent Series: The Laurent series will be mathematically derived. From Laurent, the Fourier and Taylor series are also derived.

Residues: You will be introduced to residues and how to use them to do contour integration.

Prerequisites: To take this course, you should have completed single variable Calculus, especially derivatives and integrals, and multivariable Calculus, especially line integrals and Stokes' theorem.

Original Material: This course is based on the instructor's notes on Complex Calculus, and the presentation of the results is therefore original.

Focusing on Understanding: The explanations are given by focusing on understanding and mathematically deriving the key concepts, rather than learning formulas and exercises by rote.

Benefits: Some of the results presented in this course constitute the foundations of many branches of science, including Quantum Mechanics, Quantum Field Theory, and Engineering (in the Control theory of dynamical systems, for instance). By mastering the contents of this course, you will be able to start tackling the most interesting mathematical and engineering problems.

Who this course is for: This course is suitable for anyone interested in expanding their knowledge of mathematics, including students of mathematics, physics, engineering, and related fields, as well as professionals who wish to develop their understanding of Complex Calculus.

Who this course is for:

mathematicians
physicists
engineers
computer scientists
Students interested in the concepts of Complex Calculus
Students who want to rigorously derive the concept of Fourier Series
Students who want to understand how to do contour integration

Mastering Complex Calculus: From Derivatives to Residues

What you'll learn

Explore related topics

Course content

Introduction to complex functions and derivatives of complex functions3 lectures • 49min

Integration in Complex Calculus4 lectures • 1hr 9min

Laurent Series, Fourier Series, Taylor Series5 lectures • 1hr 1min

Residues and Contour Integration15 lectures • 4hr 53min

How residues aid in the interpretation of the Fourier Transform and its inverse2 lectures • 34min

How to use complex calculus to attribute meaning to divergent series8 lectures • 1hr 28min

Some relevant properties of the Riemann Zeta function8 lectures • 1hr 13min

Requirements

Description

Who this course is for: