Advanced Integration Techniques

Advanced approaches for solving many complex integrals using special functions, some transformations and complex analysis approaches. https://web.archive.org/web/20181207122343/http://advancedintegrals.com/advanced-integration-techniques.pdf 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