Definite Integral of Algebraic, Exponential and Hyperbolic Functions Expressed in Terms of Special Functions by Robert Reynolds, Allan Stauffer

“…We evaluate several of these definite integrals of the form <inline-formula><math display="inline"><semantics><mrow><msubsup><mo>∫</mo><mrow><mn>0</mn></mrow><mo>∞</mo></msubsup><mfrac><mrow><msup><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>y</mi><mo>)</mo></mrow><mi>k</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>−</mo><mi>y</mi><mo>)</mo></mrow><mi>k</mi></msup></mrow><mrow><msup><mi>e</mi><mrow><mi>b</mi><mi>y</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow></mfrac><mi>d</mi><mi>y</mi></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><msubsup><mo>∫</mo><mrow><mn>0</mn></mrow><mo>∞</mo></msubsup><mfrac><mrow><msup><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>y</mi><mo>)</mo></mrow><mi>k</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>−</mo><mi>y</mi><mo>)</mo></mrow><mi>k</mi></msup></mrow><mrow><msup><mi>e</mi><mrow><mi>b</mi><mi>y</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mi>d</mi><mi>y</mi></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><msubsup><mo>∫</mo><mrow><mn>0</mn></mrow><mo>∞</mo></msubsup><mfrac><mrow><msup><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>y</mi><mo>)</mo></mrow><mi>k</mi></msup><mo>−</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>−</mo><mi>y</mi><mo>)</mo></mrow><mi>k</mi></msup></mrow><mrow><mo form="prefix">sinh</mo><mo>(</mo><mi>b</mi><mi>y</mi><mo>)</mo></mrow></mfrac><mi>d</mi><mi>y</mi></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msubsup><mo>∫</mo><mrow><mn>0</mn></mrow><mo>∞</mo></msubsup><mfrac><mrow><msup><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>y</mi><mo>)</mo></mrow><mi>k</mi></msup><mo>+</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>−</mo><mi>y</mi><mo>)</mo></mrow><mi>k</mi></msup></mrow><mrow><mo form="prefix">cosh</mo><mo>(</mo><mi>b</mi><mi>y</mi><mo>)</mo></mrow></mfrac><mi>d</mi><mi>y</mi></mrow></semantics></math></inline-formula> in terms of a special function where <i>k</i>, <i>a</i> and <i>b</i> are arbitrary complex numbers.…”
Get full text

Derivation of Logarithmic and Logarithmic Hyperbolic Tangent Integrals Expressed in Terms of Special Functions by Robert Reynolds, Allan Stauffer

“…We evaluate several of these definite integrals of the form <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mo>∞</mo> </msubsup> <mo form="prefix">log</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>±</mo> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>α</mi> <mi>y</mi> </mrow> </msup> <mo>)</mo> </mrow> <mi>R</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>y</mi> </mrow> </semantics> </math> </inline-formula> in terms of a special function, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>R</mi> <mo>(</mo> <mi>k</mi> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is a general function and <i>k</i>, <i>a</i> and <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula> are arbitrary complex numbers, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>R</mi> <mi>e</mi> <mo>(</mo> <mi>α</mi> <mo>)</mo> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>.…”
Get full text

The Logarithmic Transform of a Polynomial Function Expressed in Terms of the Lerch Function by Robert Reynolds, Allan Stauffer

“…This is a collection of definite integrals involving the logarithmic and polynomial functions in terms of special functions and fundamental constants. All the results in this work are new.…”
Get full text

A Quadruple Definite Integral Expressed in Terms of the Lerch Function by Robert Reynolds, Allan Stauffer

“…Special cases of this integral are evaluated in terms of special functions and fundamental constants. Almost all Lerch functions have an asymmetrical zero-distribution. …”
Get full text

Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function by Robert Reynolds, Allan Stauffer

“…We then evaluate this formula to derive new series in terms of special functions and fundamental constants. All the results in this work are new.…”
Get full text

Definite Integral Involving Rational Functions of Powers and Exponentials Expressed in Terms of the Lerch Function by Robert Reynolds, Allan Stauffer

“…This paper gives new integrals related to a class of special functions. This paper also showcases the derivation of definite integrals involving the quotient of functions with powers and the exponential function expressed in terms of the Lerch function and special cases involving fundamental constants. …”
Get full text

A Quadruple Integral Involving Product of the Struve <i><b>H</b>_v</i>(<i>β</i><i>t</i>) and Parabolic Cylinder <i>D_u</i>(<i>α</i><i>x</i>) Functions by Robert Reynolds, Allan Stauffer

“…Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distributionSpecial cases in terms fundamental constants and other special functions are produced. All the results in the work are new.…”
Get full text

Table in Gradshteyn and Ryzhik: Derivation of Definite Integrals of a Hyperbolic Function by Robert Reynolds, Allan Stauffer

“…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. …”
Get full text

A Note on Some Definite Integrals of Arthur Erdélyi and George Watson by Robert Reynolds, Allan Stauffer

“…We derive special cases of these integrals in terms of special functions not found in current literature. Special functions have the property of analytic continuation, which widens the range of computation of the variables involved.…”
Get full text

Double Integral of the Product of the Exponential of an Exponential Function and a Polynomial Expressed in Terms of the Lerch Function by Robert Reynolds, Allan Stauffer

“…This integral formula is then used to derive closed solutions in terms of fundamental constants and special functions. Almost all Lerch functions have an asymmetrical zero distribution. …”
Get full text