scholarly journals On an integral involving the logarithm function

2020 ◽  
Author(s):  
Sumit Kumar Jha
Keyword(s):  
Zeta Function ◽  
Gamma Function ◽  
Hurwitz Zeta Function ◽  
Digamma Function ◽  
Logarithm Function ◽  
Master Theorem

We use the Ramanujan's master theorem to evaluate the integral $$\int_{0}^{\infty}\frac{x^{l-1}}{(1+x)^{m+1}}\log^{n}(1+x)\, dx $$ in terms of the digamma function, the gamma function, and the Hurwitz zeta function.

scholarly journals Error bounds for the asymptotic expansion of the Hurwitz zeta function

2017 ◽  
Vol 473 (2203) ◽  
pp. 20170363 ◽  
Author(s):  
G. Nemes
Keyword(s):  
Asymptotic Expansion ◽  
Zeta Function ◽  
Gamma Function ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Remainder Term ◽  
Hurwitz Zeta Function ◽  
Polygamma Functions

In this paper, we reconsider the large- a asymptotic expansion of the Hurwitz zeta function ζ ( s , a ). New representations for the remainder term of the asymptotic expansion are found and used to obtain sharp and realistic error bounds. Applications to the asymptotic expansions of the polygamma functions, the gamma function, the Barnes G -function and the s -derivative of the Hurwitz zeta function ζ ( s , a ) are provided. A detailed discussion on the sharpness of our error bounds is also given.

scholarly journals Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1952
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Gamma Function ◽  
Integral Method ◽  
Special Functions ◽  
Incomplete Gamma Function ◽  
Hurwitz Zeta Function ◽  
Contour Integral ◽  
Fundamental Constants ◽  
Infinite Sum

We apply our simultaneous contour integral method to an infinite sum in Prudnikov et al. and use it to derive the infinite sum of the Incomplete gamma function in terms of the Hurwitz zeta function. 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.

On multiple Hurwitz zeta function of Mordell–Tornheim type

2021 ◽  
pp. 1-34
Author(s):  
Kazuhiro Onodera
Keyword(s):  
Zeta Function ◽  
Hurwitz Zeta Function ◽  
Multiple Zeta Function ◽  
Multiple Zeta ◽  
Second Derivatives ◽  
Positive Integers

We introduce a certain multiple Hurwitz zeta function as a generalization of the Mordell–Tornheim multiple zeta function, and study its analytic properties. In particular, we evaluate the values of the function and its first and second derivatives at non-positive integers.

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