Methods of contour integration

Physics

Analytic function

Calculus (dental)

Trigonometry

Mathematics

Logarithm

Evolutionary biology

Biology

Integration by parts

Improper integral

Mathematical analysis

Function (biology)

Trigonometric functions

Eigenvalues and eigenvectors

Composite material

Integral equation

Materials science

Dentistry

Programming language

Range (aeronautics)

Geometry

Constant (computer programming)

Computer science

Positive-definite matrix

Special functions

Quantum mechanics

Trigonometric integral

Applied mathematics

Continuation

Singular integral

Inverse trigonometric functions

Medicine

International Mathematical Forum

We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants such as Catalan's constant and $\pi$.

A method for evaluating definite integrals in terms of special functions with examples