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Mathematics
Complex Analysis
1. Foundations of Complex Numbers
2. Complex Functions
3. Differentiation and Analytic Functions
4. Elementary Complex Functions
5. Complex Integration
6. Consequences of Cauchy's Theorems
7. Series and Singularities
8. Residue Theory
9. Conformal Mapping
10. Advanced Topics
8.
Residue Theory
8.1.
The Residue
8.1.1.
Definition and Calculation
8.1.1.1.
Residue at Simple Poles
8.1.1.2.
Residue at Higher Order Poles
8.1.1.3.
Residue at Essential Singularities
8.1.2.
Cauchy's Residue Theorem
8.1.2.1.
Statement and Proof
8.1.2.2.
Applications to Contour Integration
8.2.
Evaluation of Real Integrals
8.2.1.
Rational Functions over Real Line
8.2.1.1.
Semicircular Contours
8.2.1.2.
Conditions for Convergence
8.2.2.
Trigonometric Integrals
8.2.2.1.
Unit Circle Method
8.2.2.2.
Fourier-Type Integrals
8.2.3.
Integrals with Branch Points
8.2.3.1.
Keyhole Contours
8.2.3.2.
Branch Cut Integration
8.3.
The Argument Principle
8.3.1.
Statement and Applications
8.3.2.
Counting Zeros and Poles
8.3.3.
Rouché's Theorem
8.3.3.1.
Statement and Applications
8.3.3.2.
Location of Zeros
8.4.
Summation of Series
8.4.1.
Residue Method for Series
8.4.2.
Mittag-Leffler's Theorem
8.4.3.
Partial Fraction Expansions
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9. Conformal Mapping