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Elementary Functions

Authored by: Yu. A. Brychkov , O. I. Marichev , N. V. Savischenko

Handbook of Mellin Transforms

Print publication date:  October  2018
Online publication date:  October  2018

Print ISBN: 9781138353350
eBook ISBN: 9780429434259
Adobe ISBN:

10.1201/9780429434259-2

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Abstract

More formulas can be obtained from the corresponding sections due to the relations a r - x r + α $$ \begin{aligned} \frac{1}{\sqrt{z+1}+1}=\frac{1}{2}{\,}_2F_1\genfrac(){0.0pt}0{\frac{1}{2},{\,}1}{2;\,-z},\quad \frac{1}{\sqrt{\sqrt{z+1}+1}}=\frac{1}{\sqrt{2}}{\,}_2F_1\genfrac(){0.0pt}0{\frac{1}{4},{\,}\frac{3}{4}}{\frac{3}{2};\,-z},\\ \frac{1}{\sqrt{1-\sqrt{z}}}+\frac{1}{\sqrt{1+\sqrt{z}}}=2 {\,}_2F_1\genfrac(){0.0pt}0{\frac{1}{4},{\,}\frac{3}{4}}{\frac{1}{2};{\,}z},\quad \dfrac{1}{\left(1-\sqrt{z}\right)^{3/2}}+\dfrac{1}{\left(1+\sqrt{z}\right)^{3/2}}=2{\,}_2F_1\genfrac(){0.0pt}0{\frac{3}{4},{\,}\frac{5}{4}}{\frac{1}{2};{\,}z},\\ \left(z+1\right)^a={\,}_1F_0\genfrac(){0.0pt}0{-a}{-z}={\,}_2F_1\genfrac(){0.0pt}0{-a,{\,}b}{b;\,-z},\quad {\,}\left(z+1\right)^a=\frac{1}{\Gamma \left(-a\right)}{\,}G_{11}^{11}\left(z{\,}\biggl |{\,}\frac{a+1}{0}\right),\\ \frac{1}{1-z}=\pi {\,}G_{22}^{11}\left(z{\,}\biggl |{\,}\frac{0,{\,}1/2}{0,{\,}1/2}\right),\quad \left(1-x\right)_+^{\alpha -1}=\Gamma \left(\alpha \right)G_{11}^{10}\left(x{\,}\biggl |{\,}\frac{\alpha }{0}\right),\quad \left(x-1\right)_+^{\alpha -1}=\Gamma \left(\alpha \right)G_{11}^{01}\left(x{\,}\biggl |{\,}\frac{\alpha }{0}\right). \end{aligned} $$  

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