Mastering Complex Calculus: From Derivatives to Residues

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Emanuele Pesaresi

9:53:37

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1. Introduction to complex functions and derivatives of complex functions

1. functions of a complex variable part 1.mp4

08:55

2. functions of a complex variable part 2.mp4

20:20

3. Concept of derivative in complex calculus.mp4

19:53

2. Integration in Complex Calculus

1. integrals of complex functions and Cauchy theorem.mp4

19:01

2. Extension of Cauchy theorem.mp4

19:22

3. Cauchy integral formula part 1.mp4

21:07

4. Cauchy integral formula part 2.mp4

09:18

3. Laurent Series, Fourier Series, Taylor Series

1. Laurent series.mp4

26:03

2. Laurent series in compact form.mp4

08:29

3. Fourier series derivation from Laurent series.mp4

14:56

4. Fourier series generalization to any period T.mp4

05:13

5. Taylor series derivation from Laurent series.mp4

06:13

4. Residues and Contour Integration

1. Concept of Residue.mp4

07:03

2. Residue Theorem.mp4

10:39

3. Calculation of residues and coefficients of the Laurent series.mp4

22:56

4. Evaluation of a real integral using complex integration (exercise 1).mp4

25:19

5. Contour integration to evaluate a real integral (exercise 2).mp4

30:54

6. Contour integration to evaluate a real integral (exercise 3).mp4

25:14

7. Another integral evaluated using the results of Complex Calculus (exercise 4).mp4

07:36

8. Contour integration to evaluate a complex integral (exercise 5).mp4

27:02

9. Another contour integration of a real integral - Exercise 6.mp4

15:36

10. Fresnel integral over the real line (formally derived with the residue theorem).mp4

14:19

11. Hilbert transform and its geometric meaning.mp4

15:42

12. Solution to the diffusion equation using complex calculus and Laplace transform.mp4

13:26

13. Representation of the Dirac Delta.mp4

32:11

14. Abel-Plana formula in complex Calculus.mp4

17:34

15. Convolution of sinc functions using complex calculus.mp4

27:06

5. How residues aid in the interpretation of the Fourier Transform and its inverse

1. The importance of the Dirac Delta in defining the Inverse Fourier Transform.mp4

16:17

2. Another integral representation of the Dirac Delta.mp4

17:27

6. How to use complex calculus to attribute meaning to divergent series

1. Complex calculus to evaluate divergent series.mp4

39:09

2. Introduction to the analytic continuation of the Riemann zeta function.mp4

02:27

3. Train of impulses expanded in a Fourier series.mp4

08:42

4. Poisson summation formula.mp4

04:48

5. Application of Poisson summation formula.mp4

06:55

6. Another representation of the Riemann zeta function.mp4

05:15

7. Functional equation of the Riemann zeta function.mp4

13:53

8. Evaluation of the Riemann zeta function at s=-3.mp4

07:17

Description

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

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

Who is this 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

What You Need to Know?

Single Variable Calculus (especially: derivatives, integrals)

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

More details

Description
Complex Calculus is an essential course that 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 tackle 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

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I obtained my PhD in "Mechanics and Advanced Engineering Sciences" in 2021.I attained a Bachelor of Science and Master of Science in Mechanical engineering in 2015 and 2017 respectively, with honors from the University of Bologna.I was the teaching tutor for the course of Mechanics of Machines from the academic year 2018 until the end of 2021 at the University of Bologna (branch of Forlì).My passion for mathematics, physics and teaching has motivated me to lecture high school and university students.My approach as a teacher is to prove to students that memory is less important for an engineer, mathematician, or physicist, than learning how to tackle a problem through logical reasoning. I believe that a teacher of scientific subjects should try to develop his students’ curiosity about the subject, rather than just concentrating on acquisition of knowledge, however important that may also be. Students should be encouraged to dig deeper and build on their knowledge by continually questioning it, rather than accepting everything at face value without a thorough understanding.For enquiries (e.g. about tutoring, or advice related to the subjects spanned by my courses), you can either contact me on LinkedIn, or you can post questions in my courses' message boards, or you can also contact me via email or on my website.You can also find the updated versions of my courses on my website.

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