# 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. : https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=7752b211-92e3-4334-9499-ad8100caa131 Recordings(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 : paragraphs 13.1 and 13.2 from the book. References: https://www.livescience.com/51399-eulers-identity.htmlA note on Euler's identity | Introduction to Laplace transforms and operational calculus. Definition of Laplace transform. Existence and uniqueness of Laplace transforms. : https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=8ce06b98-fd08-4ba5-8d8d-ad8100caf91d Recording: paragraph 6.1 from the book.References | complex_1.pdf laplace_1.pdf |

Friday 13/8 | Roots of a complex number. Remarks on the concepts on sets in the complex plane. : https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=b50f95a6-d9d0-40ec-ac59-ad8100caf967 Recording: paragraphs 13.2 and 13.3 from the book. References | Linearity of the Laplace transfrom and examples. s-shifting and examples. Transforms of derivatives. : 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 Recordings: paragraphs 6.1 and 6.2 from the book. References | 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. : 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 Recordings: paragraph 13.3 from the book. References | Transforms of derivatives and integrals with examples. ODEs. : https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=540c9146-29ba-4bc7-a29d-ad87011ae838 Recordings: paragraph 6.2 from the book.References | complex_3.pdf laplace_3.pdf ex_circuits.pdf. |

Thursday 19/8 | Analytic functions and Cauchy-Riemann equations. : https://ntnu.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=ce555dff-ed39-4811-b05c-ad87011ae876 Recordings: paragraphs 13.3 and 13.4 from the book. References | ODEs and related Initial Value Problems. Unit Step function (Heaviside function) and Second shifting theorem (t-shifting). : 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 Recordings: paragraphs 6.2 and 6.3 from the book. References | complex_4.pdf laplace_4.pdf |

Friday 20/8 | Laplace’s Equation. Harmonic Functions. Exponential function and logarithm. Hints on trigonometric functions. : 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 Recordings: paragraphs 13.4, 13.5, 13.6 and 13.7 from the bookReferences | Impulsive forces and Dirac’s Delta Function. Convolution. Hints on differentiation and integration of transforms. : Recordingsmissing : paragraphs 6.4, 6.5 and 6.6 from the book. References | complex_5.pdf laplace_5.pdf |