# Category:Complex Analysis

From ProofWiki

This category contains results about Complex Analysis.

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

**Complex analysis** is a branch of mathematics that studies complex functions.

## Subcategories

This category has the following 47 subcategories, out of 47 total.

### A

### B

### C

### D

### E

### F

### I

### J

### L

### N

### P

### R

### S

### T

### U

## Pages in category "Complex Analysis"

The following 145 pages are in this category, out of 145 total.

### A

- Absolute Value of Complex Integral
- Alternative Differentiability Condition
- Alternative Differentiability Condition/Proof 2
- Area of Parallelogram in Complex Plane
- Argument of Product equals Sum of Arguments
- Argument of Quotient equals Difference of Arguments
- Argument Principle
- Ax-Grothendieck Theorem

### B

### C

- Cauchy Integral Formula
- Cauchy's Integral Formula
- Cauchy's Integral Formula/General Result
- Cauchy's Residue Theorem
- Cauchy-Goursat Theorem
- Cauchy-Goursat Theorem/Example
- Cauchy-Hadamard Theorem
- Cauchy-Riemann Equations
- Cauchy-Riemann Equations/Expression of Derivative
- Cauchy-Riemann Equations/Necessary Condition
- Cauchy-Riemann Equations/Sufficient Condition
- Cauchy-Schwarz Inequality/Complex Numbers
- Character of Representations over C are Algebraic Integers
- Combination Theorem for Complex Derivatives
- Combination Theorem for Complex Derivatives/Combined Sum Rule
- Combination Theorem for Complex Derivatives/Multiple Rule
- Combination Theorem for Complex Derivatives/Product Rule
- Combination Theorem for Complex Derivatives/Quotient Rule
- Combination Theorem for Complex Derivatives/Sum Rule
- Complement of Closed Set in Complex Plane is Open
- Complement of Open Set in Complex Plane is Closed
- Complex Cross Product Distributes over Addition
- Complex Cross Product in Exponential Form
- Complex Dot Product in Exponential Form
- Complex Integration of Function with Primitive
- Complex Multiplication as Geometrical Transformation
- Complex Multiplication as Geometrical Transformation/Corollary
- Complex Numbers are Parallel iff Cross Product is Zero
- Complex Numbers are Perpendicular iff Dot Product is Zero
- Complex Numbers form Vector Space
- Complex Plane is Metric Space
- Complex Power by Complex Exponential is Analytic
- Complex Sequence is Cauchy iff Convergent
- Complex-Differentiable Function is Continuous
- Composite of Continuous Mappings is Continuous/Corollary
- Condition for Collinearity of Points in Complex Plane
- Condition for Points in Complex Plane to form Parallelogram
- Connected Domain is Connected by Staircase Contours
- Continuous Complex Function is Complex Riemann Integrable
- Convergence of Complex Sequence in Polar Form
- Convergence of Complex Sequence in Polar Form/Corollary
- Cross Product is Anticommutative/Complex

### D

- De Moivre's Formula
- Derivative of Complex Composite Function
- Derivative of Complex Power Series
- Derivative of Complex Power Series/Proof 2/Lemma
- Derivative of Infinite Product of Analytic Functions
- Derivative of Sequence of Holomorphic Functions
- Derivative of Uniform Limit of Analytic Functions
- Derivative of Uniform Limit of Holomorphic Functions
- Directed Smooth Curve Relation is Equivalence

### E

- Equation for Line through Two Points in Complex Plane
- Equation of Unit Circle in Complex Plane
- Equivalence of Definitions of Absolute Convergence of Product of Complex Numbers
- Equivalence of Definitions of Analytic Function
- Equivalence of Definitions of Complex Cross Product
- Equivalence of Definitions of Complex Dot Product
- Equivalence of Definitions of Exterior Point (Complex Analysis)
- Equivalence of Definitions of Open Set (Complex Analysis)
- Equivalence of Local Uniform Convergence and Compact Convergence
- Euler's Identity
- Existence of Laurent Series

### G

### H

### L

- Lemniscate of Bernoulli as Locus in Complex Plane
- Limit of Function Unique
- Limit Point of Set in Complex Plane not Element is Boundary Point
- Limits of Real and Imaginary Parts
- Linear Combination of Complex Integrals
- Linearly Independent over the Rational Numbers iff Linearly Independent over the Integers
- Liouville's Theorem (Complex Analysis)
- Little Picard Theorem
- Logarithmic Derivative of Infinite Product of Analytic Functions
- Logarithmic Derivative of Infinite Product of Holomorphic Functions
- Logarithmic Derivative of Product of Analytic Functions

### M

### P

### R

- Radius of Convergence of Derivative of Complex Power Series
- Reciprocal of Holomorphic Function
- Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints
- Reparameterization of Directed Smooth Curve Preserves Image
- Residue at Simple Pole
- Residue of Quotient
- Residue Theorem
- Reversed Directed Smooth Curve is Directed Smooth Curve
- Riemann Removable Singularities Theorem
- Roots of Complex Number
- Roots of Unity lie on Vertices of Regular Polygon
- Roots of Unity under Multiplication form Cyclic Group
- Rouché's Theorem