# 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

10.1201/9780429434259-2

#### 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}

## Use of cookies on this website

We are using cookies to provide statistics that help us give you the best experience of our site. You can find out more in our Privacy Policy. By continuing to use the site you are agreeing to our use of cookies.