# Gaussian hypergeometric function Calculator

## Calculates Gaussian hypergeometric function 2F1(a,b;c;z).

 $\normal Hypergeometric\ function\ of\ the\ 1st\ kind\\[15]\large {}_2F_1(a,b;c;z)=1+\frac{ab}{c}z+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{z^2}{2!}+\cdots\\\hspace{90}=\sum_{\small n=0}^{\small \infty}\frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!}\\[10]$
 a b c z 6digit10digit14digit18digit22digit26digit30digit34digit38digit42digit46digit50digit 2F1(a,b;c;z)
 $\normal Hypergeometric\ differential\ equation\\[15](1)\ z(1-z)y''+(c-(a+b+1)z))y'-aby=0\\\hspace{25} y={}_2F_1(a,b;c;z)\\[10](2)\ {}_2F_1(a,b;c;z)=1+{\large\frac{ab}{c}}z+{\large\frac{a(a+1)b(b+1)}{c(c+1)}\frac{z^2}{2!}}+\cdots\\\hspace{90}={\large \sum_{\small n=0}^{\small \infty}}{\large\frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!}}\\[10]$
Gaussian hypergeometric function
 [1-10] /20 Disp-Num5103050100200
[1]  2018/05/25 01:14   Male / 40 years old level / An engineer / Very /
Purpose of use
For checking an expression of the binomial-binomial convolution probability mass function. Used to tabulate some values (too much work to get my own implementation of the Gauss'' hypergeometric function for that simple purpose!)
Comment/Request
An option to avoid commas in the result would be appreciated (for direct copy-paste)
[2]  2018/04/30 21:11   Male / 50 years old level / A teacher / A researcher / A little /
Purpose of use
research
[3]  2018/03/16 08:08   Female / 20 years old level / High-school/ University/ Grad student / Useful /
Purpose of use
Use WKB method to calculate eigenvalues of harmonic oscillator
[4]  2017/09/11 13:59   Male / 40 years old level / An engineer / Very /
Purpose of use
Solve integral
Comment/Request
Your calculator is extremely helpful. Thank you!
Which series formula are you using for the hypergeometric fucntion 2F1(a,b;c;z) in case of z<0, but |z|>1, for example z=-2?
from Keisan
We use "z -> z/z-1" and "z -> 1/z" convert formula in case of z<0 and |z|>1.
[5]  2017/09/09 09:46   Male / 40 years old level / An engineer / Useful /
Purpose of use
Solve a integral problem via hypergeometric summation
[6]  2016/08/26 21:52   Male / 30 years old level / A teacher / A researcher / Very /
Purpose of use
Trying to write a code for computation of bessel function for complex argument. I will love to have the algorithm you implemented
[7]  2016/06/05 07:58   Male / 20 years old level / An office worker / A public employee / Very /
Purpose of use
to test my own implementation
Bug report
a=0.1; b=0.2; c=0.4; z=2
Keisan result: 1.956835345165 -0.4667180917933i
My result: 1.01537811446162-0.16082009437679i; not claiming it is 100% correct, just trying to implement this: http://arxiv.org/pdf/0708.0116v2.pdf

Proof of error:
Keisan (b=0.1999999999999999): 1.0153781144616367 -0.1608200943767892i
Keisan (b=0.2000000000000001): 1.0153781144616367 -0.1608200943767894i

Is there anything specific about 2F1 function around b=0.2?
from Keisan
We fixed a bug about Gaussian hypergeometric function. Thank you for your advice.
[8]  2015/04/23 07:03   Male / 60 years old level or over / A retired people / Useful /
Purpose of use
Solve a integral problem via hypergeometric summation
[9]  2015/04/22 03:11   Male / 60 years old level or over / Others / A little /
Purpose of use
Checking the stability range of 2F1 for complex z-values
Comment/Request
How do you enter z-complex=values on your Gaussian Hypergeometric calculator?
[10]  2015/02/25 02:17   Male / 30 years old level / A teacher / A researcher / Useful /
Purpose of use
Verify special implementation in own code

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