A method for evaluating definite integrals in terms of special functions with examples
Robert Reynolds, Allan Stauffer
Abstract
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 $\pi$.
Topics & Concepts
MathematicsTrigonometric functionsSpecial functionsTrigonometric integralRange (aeronautics)Applied mathematicsCalculus (dental)TrigonometryMathematical analysisAnalytic functionPositive-definite matrixMethods of contour integrationImproper integralConstant (computer programming)Term (time)Simple (philosophy)Algebra over a fieldInverse trigonometric functionsIntegration using Euler's formulaFunction (biology)Constant coefficientsElementary functionOrder of integration (calculus)Numerical integrationIntegral equationMultiple integralType (biology)Integration by partsProofs of trigonometric identitiesMathematical functions and polynomialsMathematical and Computational MethodsAdvanced Mathematical Theories