Complex Analysis

5.

Complex Integration

5.1.

Contours and Paths

5.1.1.

Parametric Curves

5.1.1.1.

Smooth and Piecewise Smooth Curves

5.1.1.2.

5.1.1.3.

Closed Contours

5.1.2.

5.1.2.1.

Length of a Contour

5.1.2.2.

Rectifiable Curves

5.2.

5.2.1.

Definition of Complex Line Integral

5.2.1.1.

Integral with Respect to z

5.2.1.2.

Integral with Respect to Arc Length

5.2.2.

Properties of Line Integrals

5.2.2.1.

5.2.2.2.

5.2.2.3.

Effect of Orientation

5.2.3.

Estimation Lemma

5.2.3.1.

5.3.

Fundamental Theorems

5.3.1.

Fundamental Theorem for Line Integrals

5.3.1.1.

Antiderivatives

5.3.1.2.

Path Independence

5.3.2.

Green's Theorem in the Complex Plane

5.3.2.1.

Statement and Applications

5.4.

Cauchy's Theorem

5.4.1.

Cauchy-Goursat Theorem

5.4.1.1.

Simply Connected Domains

5.4.1.2.

5.4.2.

Deformation of Contours

5.4.2.1.

Homotopy of Paths

5.4.2.2.

Multiply Connected Domains

5.5.

Cauchy's Integral Formula

5.5.1.

Integral Formula for Function Values

5.5.1.1.

Statement and Proof

5.5.2.

Integral Formula for Derivatives

5.5.2.1.

Higher Order Derivatives

5.5.2.2.

Cauchy's Inequality

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6. Consequences of Cauchy's Theorems