Introductory course in Laplace transforms and complex analysis 2021

Lecturer: Andrea Leone.

This course provides a brief introduction to Laplace transforms and complex analysis for the incoming two year master students who don't have the necessary backgrounds or wish a repetition of the subjects. The students do not need any prior knowledge other than that provided by basic undergraduate courses in mathematics. The course is voluntary and gives no credits. There is no exam and no registration.

Lectures are planned with physical attendance and they will not be in hybrid format or online. I will try to upload notes on this site before each lecture. If I manage I will also upload recordings of the lectures on Panopto.

Syllabus: Chapter 6 (Laplace Transforms) and Chapter 13 (Complex Numbers and Functions) from:
Erwin Kreyszig
Advanced Engineering Mathematics
10th Edition
ISBN 0-470-45836-4

Timeplan: Timetable:

Thursday 12/8 Friday 13/8 Wednesday 18/8 Thursday 19/8 Friday 20/8
Room R1 R1 S2 S2 S2
13:15-14:00 Complex analysis Laplace Complex analysis Laplace Complex analysis
14:15-15:00 Complex analysis Laplace Complex analysis Laplace Complex analysis
15:15-16:00 Laplace Complex analysis Laplace Complex analysis Laplace

Lecture plan

Complex analysis Laplace transforms Notes
Thursday 12/8 Introduction to complex analysis. Complex numbers and operations on complex numbers. The geometric interpretation of complex numbers. Polar form of complex numbers. Multiplication and division in polar form. Triangle inequalities. Exponential form of a complex number. Integer powers of a complex number. De Moivre’s formula. Recordings: https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=7752b211-92e3-4334-9499-ad8100caa131 (the camera in the first part of this recording was not centered) and https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=277b3c1a-f2f6-4aea-badb-ad8100cacf41 References: paragraphs 13.1 and 13.2 from the book. A note on Euler's identity: https://www.livescience.com/51399-eulers-identity.htmlIntroduction to Laplace transforms and operational calculus. Definition of Laplace transform. Existence and uniqueness of Laplace transforms. Recording: https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=8ce06b98-fd08-4ba5-8d8d-ad8100caf91d References: paragraph 6.1 from the book.complex_1.pdf laplace_1.pdf
Friday 13/8 Roots of a complex number. Remarks on the concepts on sets in the complex plane. Recording: https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=b50f95a6-d9d0-40ec-ac59-ad8100caf967 References: paragraphs 13.2 and 13.3 from the book. Linearity of the Laplace transfrom and examples. s-shifting and examples. Transforms of derivatives. Recordings: https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=866140b1-9530-4920-8691-ad8100caa171 and https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=047e428d-165f-4323-9b19-ad8100cacf85 References: paragraphs 6.1 and 6.2 from the book. complex_2.pdf laplace_2.pdf
Wednesday 18/8 Example on the usage of complex numbers to solve a series RLC circuit. Complex functions and examples. Limit, continuity and derivative of complex functions. Recordings: https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=cc08ba26-7af2-4f68-b07b-ad87011a90ed and https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=28196ba6-0df1-479a-9a20-ad87011ab7f8 References: paragraph 13.3 from the book. Transforms of derivatives and integrals with examples. ODEs. Recordings: https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=540c9146-29ba-4bc7-a29d-ad87011ae838 References: paragraph 6.2 from the book.complex_3.pdf laplace_3.pdf ex_circuits.pdf.
Thursday 19/8 Analytic functions and Cauchy-Riemann equations. Recordings: https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=ce555dff-ed39-4811-b05c-ad87011ae876 References: paragraphs 13.3 and 13.4 from the book. ODEs and related Initial Value Problems. Unit Step function (Heaviside function) and Second shifting theorem (t-shifting). Recordings: https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=c8f1179b-c8ea-4742-85e8-ad87011a9132 and https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=3d665c81-8566-47b7-99ab-ad87011ab83d References: paragraphs 6.2 and 6.3 from the book. complex_4.pdf laplace_4.pdf
Friday 20/8 Laplace’s Equation. Harmonic Functions. Exponential function and logarithm. Hints on trigonometric functions. Recordings: https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=05cd35ea-29f2-47ba-bb07-ad87011a9147 and https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=5b6a57ca-a4be-4ff8-8d89-ad87011ab854 References: paragraphs 13.4, 13.5, 13.6 and 13.7 from the book Impulsive forces and Dirac’s Delta Function. Convolution. Hints on differentiation and integration of transforms. Recordings: missing References: paragraphs 6.4, 6.5 and 6.6 from the book. complex_5.pdf laplace_5.pdf
2021-08-21, Andrea Leone