# Category:Complex Contour Integrals

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This category contains results about **Complex Contour Integrals**.

Definitions specific to this category can be found in Definitions/Complex Contour Integrals.

Let $C$ be a contour defined by a finite sequence $C_1, \ldots, C_n$ of directed smooth curves in the complex plane $\C$.

Let $C_k$ be parameterized by the smooth path:

- $\gamma_k: \closedint {a_k} {b_k} \to \C$

for all $k \in \set {1, \ldots, n}$.

Let $f: \Img C \to \C$ be a continuous complex function, where $\Img C$ denotes the image of $C$.

The **contour integral of $f$ along $C$** is defined by:

- $\ds \int_C \map f z \rd z = \sum_{k \mathop = 1}^n \int_{a_k}^{b_k} \map f {\map {\gamma_k} t} \map {\gamma_k'} t \rd t$

## Subcategories

This category has the following 4 subcategories, out of 4 total.

### C

- Cauchy-Goursat Theorem (9 P)

### O

## Pages in category "Complex Contour Integrals"

The following 23 pages are in this category, out of 23 total.

### C

- Cauchy-Goursat Theorem
- Complex Contour Integral as Contour Integrals
- Complex Integration by Substitution
- Complex Riemann Integral is Contour Integral
- Concatenation of Contours is Contour
- Contour Integral along Reversed Contour
- Contour Integral is Independent of Parameterization
- Contour Integral is Well-Defined
- Contour Integral of Closed Contour Split into Two Contours
- Contour Integral of Concatenation of Contours