scholarly journals Definite integrals involving product of logarithmic functions and logarithm of square root functions expressed in terms of special functions

2020 ◽  
Vol 5 (6) ◽  
pp. 5724-5733
Author(s):  
Robert Reynolds ◽  
◽  
Allan Stauffer
Keyword(s):  
Special Functions ◽  
Square Root ◽  
Definite Integrals ◽  
Root Functions ◽  
Logarithmic Functions

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

2021 ◽  
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Special Function ◽  
Special Functions ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Definite Integral ◽  
Definite Integrals ◽  
Catalan’S Constant ◽  
Infinite Sums ◽  
Logarithmic Functions ◽  
Mathematical Constants

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 π

scholarly journals Definite Integrals Involving Combinations of Powers and Logarithmic Functions of Complicated Arguments Expressed in Terms of the Hurwitz Zeta Function

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Special Functions ◽  
Zeta Functions ◽  
Hurwitz Zeta Function ◽  
Fundamental Constants ◽  
Closed Formula ◽  
Definite Integrals ◽  
Logarithmic Functions

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 functions. 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.

scholarly journals The Double Laplace Transform Expressed in terms of the Lerch Transcendent

2021 ◽  
Vol 14 (3) ◽  
pp. 618-637
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Laplace Transform ◽  
Special Functions ◽  
Fundamental Constants ◽  
Definite Integrals ◽  
Special Values ◽  
Double Laplace Transform ◽  
Lerch Transcendent

In this manuscript, the authors derive a formula for the double Laplace transform expressed in terms of the Lerch Transcendent. The log term mixes the variables so that the integral is not separable except for special values of k. The method of proof follows the method used by us to evaluate single integrals. This transform is then used to derive definite integrals 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.

scholarly journals Expressions of the Laguerre polynomial and some other special functions in terms of the generalized Meijer $ G $-functions

2021 ◽  
Vol 6 (11) ◽  
pp. 11631-11641
Author(s):  
Syed Ali Haider Shah ◽  
◽  
Shahid Mubeen
Keyword(s):  
Hypergeometric Functions ◽  
Laguerre Polynomial ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Exponential Functions ◽  
Confluent Hypergeometric Functions ◽  
Logarithmic Functions

<abstract><p>In this paper, we investigate the relation of generalized Meijer $ G $-functions with some other special functions. We prove the generalized form of Laguerre polynomials, product of Laguerre polynomials with exponential functions, logarithmic functions in terms of generalized Meijer $ G $-functions. The generalized confluent hypergeometric functions and generalized tricomi confluent hypergeometric functions are also expressed in terms of the generalized Meijer $ G $-functions.</p></abstract>

scholarly journals Definite Integral of Algebraic, Exponential and Hyperbolic Functions Expressed in Terms of Special Functions

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1425
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Special Function ◽  
Special Functions ◽  
Complex Numbers ◽  
Definite Integral ◽  
Hyperbolic Functions ◽  
Closed Form Solutions ◽  
Definite Integrals ◽  
Arbitrary Complex ◽  
Famous Book

While browsing through the famous book of Bierens de Haan, we came across a table with some very interesting integrals. These integrals also appeared in the book of Gradshteyn and Ryzhik. Derivation of these integrals are not listed in the current literature to best of our knowledge. The derivation of such integrals in the book of Gradshteyn and Ryzhik in terms of closed form solutions is pertinent. We evaluate several of these definite integrals of the form ∫0∞(a+y)k−(a−y)keby−1dy, ∫0∞(a+y)k−(a−y)keby+1dy, ∫0∞(a+y)k−(a−y)ksinh(by)dy and ∫0∞(a+y)k+(a−y)kcosh(by)dy in terms of a special function where k, a and b are arbitrary complex numbers.

scholarly journals A response analysis of wheat and barley to nitrogen in Finland

1993 ◽  
Vol 2 (6) ◽  
pp. 465-479
Author(s):  
John Sumelius
Keyword(s):  
Functional Form ◽  
Yield Response ◽  
Response Analysis ◽  
Square Root ◽  
Nitrogen Application ◽  
Producer Price ◽  
Root Functions ◽  
Price Increases ◽  
Profit Maximizing ◽  
Better Than

A nonlinear Mitscherlich function was found to be superior to quadratic and square root functions in estimating yield response to nitrogen based on a Finnish sample of barley. Nonnested hypothesis testing (J-test) indicated the Mitscherlich functional form to fit the data better than the quadratic form based on this sample. In the analysis of the crop response for spring wheat the Mitscherlich functional form could not be proved superior by a J-test. The inferred profit maximizing nitrogen fertilization levels based on the Mitscherlich functional form exceeded the quadratic polynomial forms and were lower than the inferred levels using square root specifications. Implementing 100% nitrogen price increases or 50% producer price reductions lowered the profit maximizing nitrogen application doses by 20-24%, according to the Mitscherlich specification.

scholarly journals Definite Integral of Power and Algebraic Functions in terms of the Lerch Function

2021 ◽  
Vol 14 (3) ◽  
pp. 980-988
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Present Note ◽  
Special Functions ◽  
Integral Formula ◽  
Definite Integral ◽  
Algebraic Functions ◽  
Definite Integrals ◽  
Cauchy’S Integral Formula ◽  
Logarithmic Integral

Bierens de haan (1867) evaluated a definite integral involving the cotangent function and this result was also listed in Gradshteyn and Ryzhik (2007). The objective of this present note is to use this integral along with Cauchy's integral formula to derive a definite logarithmic integral in terms of the Lerch function. We will use this integral formula to produce a table of known and new results in terms of special functions and thereby expanding the list of definite integrals in both text books.

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