I know how to derive the binomial approximation for [itex](1+\alpha x)^{\gamma}[/itex] using a Mellin transform, but for [itex](1-\alpha x)^{\gamma}[/itex] the method appears to fail because I can't take x to infinity.(adsbygoogle = window.adsbygoogle || []).push({});

Here is the basics of the method. Take the Mellin transform of [itex](1+\alpha x)^{\gamma}[/itex]:

[tex]M(p) = \int^\infty_0 (1+\alpha x)^{-\gamma}x^{p-1}dx[/tex]

Do some change of variables trickery:

[tex]

\begin{align*}

M(p) &= \alpha^{-p}\int^1_0(1-z)^{\gamma-p-1}z^{p-1} dz\\

&=\alpha^{-p}\frac{\Gamma(\gamma-p)\Gamma(p)}{\Gamma(\gamma)}

\end{align*}

[/tex]

Use the inverse Mellin transform and close the integral to the left:

[tex]

\begin{align*}

(1+\alpha x)^{-\gamma}&=\frac{1}{2\pi i}\int^{c+i\infty}_{c-i\infty}(x\alpha)^{-p}\frac{\Gamma(\gamma-p)\Gamma(p)}{\Gamma(\gamma)}dp\\

&=\sum_{n=0}^{\infty}(\alpha x)^n \frac{(-1)^n}{n!}\frac{\Gamma(\gamma+n)}{\Gamma(\gamma)}

\end{align*}

[/tex]

But taking the Mellin transform of [itex](1-\alpha x)^{\gamma}[/itex] fails immediately:

[tex]M(p) = \int^\infty_0 (1-\alpha x)^{-\gamma}x^{p-1}dx[/tex]

The integrand will become complex quite quickly. I've tried playing around with changes of variables, but I can't figure it out.

Can this method be adapted for this purpose?

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Binomial approximation using Mellin transform

Loading...

Similar Threads - Binomial approximation using | Date |
---|---|

B Need some help with understanding linear approximations | Feb 17, 2018 |

Binomial coefficients sum | Dec 12, 2015 |

Quotient rule and binomial theorem | Nov 18, 2013 |

Binomial theorem in rudin | Jul 7, 2012 |

Binomial series | May 25, 2012 |

**Physics Forums - The Fusion of Science and Community**