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Allan Stauffer

Publications and source records attributed to Allan Stauffer.

8 recordsLinked to original sources
arXiv2022

A Double Chebyshev Series: Derivation And Evaluation

In this paper we use a contour integral method to derive a generating function in the form of a double series involving the product of two Chebyshev polynomials over generalized independent indices expressed in terms of the incomplete gamma function. The generating function represents a more generalized form relative to current literature. A possible application of this function to solving partial differential equations is discussed and some special cases of this generating function are derived. The work involved in the computation of this generating function is easier relative to previous methods as we have a closed form solution as opposed to numerical methods.

math.GM↗
arXiv2022

A Sextuple Integral Containing the Product of Associated Legendre polynomials $P_v^u(x) P_{ν}^{μ}(y)$: Derivation and Evaluation

In this present paper we derive a six dimensional integral containing the product of the Associated Legendre Polynomials $P_v^u(x) P_{ν}^{μ}(y)$ where the indices are different and general. Included in the kernel of this integral is the generalized logarithmic function and coefficient logarithmic functions. The derivation of this integral is written in terms of the Hurwitz-Lerch zeta function and constant coefficients raised to a power. Special cases of this integral are derived in terms of fundamental constants and other special functions. All the results in this work are new.

math.GM↗
arXiv2021

Double integral of logarithm and exponential function expressed in terms of the Lerch function

In this work the authors use their contour integral method to derive a double integral connected to the modified Bessel function of the second kind and express it in terms of the Lerch function. There are some useful results relating double integrals of certain kinds of functions to ordinary integrals for which we know no general reference. Thus a table of integral pairs is given for interested readers. The majority of the results in this work are new.

math.GM↗
arXiv2021

Definite integrals involving combinations of powers and logarithmic functions of complicated arguments expressed in terms of the Hurwitz zeta function

In this manuscript, the authors derive closed formula for definite integrals of combinations of powers and logarithmic functions of complicated arguments and express these integrals in terms of the Hurwitz zeta. These derivations are then expressed in terms of fundamental constants, elementary and special functions. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.

math.GM↗
arXiv2021

An integral's journey over the real line

In 1826 Cauchy presented an Integral over the real line. Al and I thought a derivation would be mighty fine. So we packed our contour integral bags that day, and we now present an analytic continuation this time.

math.GM↗
arXiv2021

Table in Gradshteyn and Ryzhik: Derivation of definite integrals of a Hyperbolic Function

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 $\pi$.

math.CA↗
arXiv2019

A Method for Evaluating Definite Integrals in terms of Special Functions with Examples

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 $\pi$.

math.NT↗