# Complex Calculus

## Course Description:

Holomorphic complex functions. Complex integration, Cauchy-Goursat theorem and consequences. Taylor and Laurent series and singularities. The Residue Theorem and applications in the real integral calculus. Conformal mappings and applications in Partial Differential Equations.

• Semester 8
• Teaching hours 3

Analysis I & II

### Course Units

# Title Description Hours
1 Introductory concepts Complex numbers, modulus and polar form of a complex number, complex sequences and series, basic topological concepts in the complex plane. 1Χ4=4
2 Complex functions, differentiability and holomorphy Complex functions, limit and continuity. Exponential function, logarithms and trigonometric functions. Differentiable complex functions, Cauchy-Riemann conditions, holomorphic complex functions. . 2Χ4=8
3 Complex integration Complex contour integral. Cauchy-Goursat theorem, Deformation Principle, Cauchy's Integral formulas, Liouville's theorem, Maximum Modulus Principle, harmonic functions, uniqueness of the solution to the Poisson problem . . 2Χ4=8
4 Power and Laurent series, singularities. Power series and convergence radius. Taylor's theorem and Taylor's expansions of basic complex numbers. Laurent series and isolated singularities: removable singularities, poles and essential singularities. 2Χ4+2=10
5 Residue theorem and applications Calculus of Residues and Applications in the calculation of trigonometric and improrer real integrals. 2Χ4=8
6 Conformal functions. Conformal functions. Mobius transformations and applications in Boundary Value Problems (PDE's). 1Χ4+2=6
7 Σύμμορφη απεικόνιση. Σύμμορφη απεικόνιση. Μετασχηματισμοί Mobius, θεώρημα απεικόνισης του Riemann, μετασχηματισμός Schwarz-Christoffel. Εφαρμογές της σύμμορφης απεικόνισης. 2Χ4=8

### Learning Objectives

In the end of the semester, the students will be supposed to have the following knowledge: 1.Basic concepts and results related to holomorphic functions, complex integration and conformal functions. 2.Applications of the above mentioned concepts and results in real integral calculus and PDE's.

### Teaching Methods

 Teaching methods Lectures, exercises, tests and homework Blackboard presentations

### Student Assessment

• Final written exam: 100%

### Textbooks - Bibliography

1. ΕΦΑΡΜΟΣΜΕΝΗ ΜΙΓΑΔΙΚΗ ΑΝΑΛΥΣΗ ΚΡΑΒΒΑΡΙΤΗΣ Δ. ΤΣΟΤΡΑΣ 2016

2. ΜΕΡΙΚΕΣ ΔΙΑΦΟΡΙΚΕΣ ΕΞΙΣΩΣΕΙΣ & ΜΙΓΑΔΙΚΕΣ ΣΥΝΑΡΤΗΣΕΙΣ: ΘΕΩΡΙΑ ΚΑΙ ΕΦΑΡΜΟΓΕΣ Ν.ΣΤΑΥΡΑΚΑΚΗΣ Ν. ΣΤΑΥΡΑΚΑΚΗΣ (1η ΕΚΔΟΣΗ) 2016

3. ΕΠΙΛΕΚΤΑ ΘΕΜΑΤΑ ΜΙΓΑΔΙΚΗΣ ΑΝΑΛΥΣΗΣ. ΣΥΜΜΟΡΦΗ ΑΠΕΙΚΟΝΙΣΗ. ΕΦΑΡΜΟΓΕΣ ΤΗΣ ΜΙΓΑΔΙΚΗΣ ΑΝΑΛΥΣΗΣ ΣΤΗ ΘΕΩΡΙΑ ΠΕΔΙΩΝ (ΤΕΥΧΗ ΤΡΙΑ) ΧΑΪΝΗΣ Ι. ΦΟΥΝΤΑΣ 2005

4. ΜΙΓΑΔΙΚΗ ΑΝΑΛΥΣΗ BAK J. ,NEWMAN D.J. LEADER BOOKS 2004

5. ΒΑΣΙΚΗ ΜΙΓΑΔΙΚΗ ΑΝΑΛΥΣΗ MARSDEN J.HOFFMAN M. ΣΥΜΜΕΤΡΙΑ 1994

## Lecture Time - Place:

• Monday, 09:45 – 12:30,
Rooms:
• Ζ. Κτ. 1 Πολ., Αιθ. 15

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