Advanced Integration Techniques

Differentiation under the integral sign
Example
Example
Example
Laplace Transform
Basic Introduction
Example
Example
Convolution
Inverse Laplace transform
Example
Interesting results
Example
Example
Example
Gamma Function
Definition
Example
Example
Exercises
Extension
Theorem
Reduction formula
Other Representations
Euler Representation
Example
Weierstrass Representation
Laurent expansion
Example
More values
Legendre Duplication Formula
Example
Euler's Reflection Formula
Example
Example
Beta Function
Representations
First integral formula
Second integral formula
Geometric representation
Example
Example
Example
Example
Example
Example
Example
Example
Exercise
Digamma function
Definition
Example
Difference formulas
First difference formula
Second difference formula
Example
Series Representation
Some Values
Example
Integral representations
First Integral representation
Second Integral representation
Third Integral representation
Fourth Integral representation
Gauss Digamma theorem
More results
Example
Example
Example
Example
Example
Zeta function
Definition
Bernoulli numbers
Relation between zeta and Bernoulli numbers
Exercise
Integral representation
Hurwitz zeta and polygamma functions
Definition
Relation between zeta and polygamma
Example
Dirichlet eta function
Definition
Relation to Zeta function
Integral representation
Polylogarithm
Definition
Relation to other functions
Integral representation
Square formula
Exercise
Dilogarithms
Definition
First functional equation
Second functional equation
Third functional equation
Example
Example
Example
Example
Ordinary Hypergeometric function
Definition
Some expansions using the hypergeomtric function
Exercise
Integral representation
Transformations
Special values
Error Function
Definition
Complementary error function
Imaginary error function
Properties
Relation to other functions
Example
Example
Example
Example
Exercise
Exponential integral function
Definition
Example
Example
Example
Example
Example
Exercise
Complete Elliptic Integral
Complete elliptic of first kind
Complete elliptic of second kind
Hypergeometric representation
Example
Identities
Special values
Differentiation of elliptic integrals
Euler sums
Definition
Generating function
Integral representation of Harmonic numbers
Example
Example
General formula
Example
Example
Example
Relation to polygamma
Integral representation for r=1
Symmetric formula
Example
Sine Integral function
Definition
Example
Example
Example
Example
Example
Example
Example
Cosine Integral function
Definition
Relation to Euler constant
Example
Example
Example
Example
Example
Example
Integrals involving Cosine and Sine Integrals
Example
Example
Logarithm Integral function
Definition
Example
Find the integral
Find the integral
Example
Example
Clausen functions
Definition
Duplication formula
Example
Example
Clausen Integral function
Definiton
Integral representation
Duplication formula
Example
Example
Example
Second Integral representation
Example
Barnes G function
Definition
Functional equation
Reflection formula
Values at positive integers
Relation to Hyperfactorial function
Loggamma integral
Glaisher-Kinkelin constant
Relation to Glaisher-Kinkelin constant
Example
Example
Relation to Howrtiz zeta function
Example
Complex Analysis
Introduction to complex numbers
Polar representation
Complex functions
Exponential function
Sine and Cosine and hyperbolic functions
Complex logarithm
Taylor and Laurent expansions
Poles and residues
Integration around paths
Bounds on integrals
Contours around poles
Real integrals using contour integration
Trigonometric functions
Example
Integrating around an ellipse
Example
Creating crazy integrals
Example
Example
Example
Trigonometric functions with rationals of polynomials
Example
Integration along contours with detours
Example
Integrals of functions with branch cuts
Example
Example
Example
Example
Example
Example
Rectangular contours
Example
Example
Example
Triangular contours
Example
Residue at infinity
Example
Inverse of Laplace transform
Example
Infinite sums
Example