HYPERG(3pm)

✨ HYPERG(3pm) ✨

okay bestie so you opened the HYPERG(3pm) manual page and it's giving math girlie energy, so let's get into it 💖

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🌸 NAME 🌸

PDL::GSLSF::HYPERG — it's the PDL interface to GSL Special Functions, and yes that's a lot of capital letters but we move 💅

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🦋 DESCRIPTION 🦋

So basically this is an interface to the Special Function package that lives inside the GNU Scientific Library (aka GSL, aka the icon). That's the whole description, she's brief, she's mysterious, we respect her 👑

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🎀 FUNCTIONS 🎀

🌟 gsl_sf_hyperg_0F1

Signature goes a lil something like this:

Signature: (double x(); double [o]y(); double [o]e(); double c)

and then the original page drops this whole comment block, so here it is verbatim because it's literally the math:

/* Hypergeometric function related to Bessel functions 0F1[c,x] =
Gamma[c]    x^(1/2(1-c)) I_{c-1}(2 Sqrt[x]) Gamma[c] (-x)^(1/2(1-c))
J_{c-1}(2 Sqrt[-x])

ngl, gsl_sf_hyperg_0F1 does not process bad values. The way it works: if the bad-value flag is set for any of the input ndarrays, it will set the bad-value flag of all the output ndarrays. That's the deal, no cap 🫶

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🌟 gsl_sf_hyperg_1F1

Signature:

Signature: (double x(); double [o]y(); double [o]e(); double a; double b)

Confluent hypergeometric function for integer parameters, and the vibe is 1F1[a,b,x] = M(a,b,x) ✨

fr fr, gsl_sf_hyperg_1F1 does not process bad values either. Bad-value flag set on any input ndarray → bad-value flag set on all output ndarrays. Every time 💀

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🌟 gsl_sf_hyperg_U

Signature:

Signature: (double x(); double [o]y(); double [o]e(); double a; double b)

Confluent hypergeometric function for integer parameters, and it's just U(a,b,x) 🧚

gsl_sf_hyperg_U does not process bad values. Same rule as always: flag set on any input ndarray means flag set on all output ndarrays 🍓

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🌟 gsl_sf_hyperg_2F1

Signature:

Signature: (double x(); double [o]y(); double [o]e(); double a; double b; double c)

Confluent hypergeometric function for integer parameters, and it's 2F1[a,b,c,x] 💜

gsl_sf_hyperg_2F1 does not process bad values. Any input ndarray has the flag set → all output ndarrays get the flag set. iconic behavior honestly 🌈

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🌟 gsl_sf_hyperg_2F1_conj

Signature:

Signature: (double x(); double [o]y(); double [o]e(); double a; double b; double c)

Gauss hypergeometric function, and she's 2F1[aR + I aI, aR - I aI, c, x] 👑

gsl_sf_hyperg_2F1_conj does not process bad values. Bad-value flag set on any input ndarray → set on all output ndarrays, periodt 💅

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🌟 gsl_sf_hyperg_2F1_renorm

Signature:

Signature: (double x(); double [o]y(); double [o]e(); double a; double b; double c)

Renormalized Gauss hypergeometric function: 2F1[a,b,c,x] / Gamma[c] ✨

gsl_sf_hyperg_2F1_renorm does not process bad values. Flag set on any input ndarray → flag set on all output ndarrays 🦋

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🌟 gsl_sf_hyperg_2F1_conj_renorm

Signature:

Signature: (double x(); double [o]y(); double [o]e(); double a; double b; double c)

Renormalized Gauss hypergeometric function, and the formula is 2F1[aR + I aI, aR - I aI, c, x] / Gamma[c] 💖

gsl_sf_hyperg_2F1_conj_renorm does not process bad values. Any input ndarray with the flag set → all output ndarrays with the flag set. You know the drill by now 🙈

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🌟 gsl_sf_hyperg_2F0

Signature:

Signature: (double x(); double [o]y(); double [o]e(); double a; double b)

Mysterious hypergeometric function. 😭 and here's the tea: the series representation is a divergent hypergeometric series. However, for x < 0 we have:

2F0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)

so it's not all chaos, there's a little escape hatch for negative x 🌸

gsl_sf_hyperg_2F0 does not process bad values. If the flag is set for any of the input ndarrays, it sets the bad-value flag of all output ndarrays. no notes 💜

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👑 AUTHOR 👑

This file is copyright (C) 1999 Christian Pellegrin <chri@infis.univ.trieste.it> — All rights reserved. There is no warranty. You are allowed to redistribute this software / documentation under certain conditions. For the details, go look at the file COPYING in the PDL distribution. And if this file ever gets separated from the PDL distribution, the copyright notice should be included in the file. 🫶

Also, the GSL SF modules were written by G. Jungman. Legends only ⭐

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and that's the whole page bestie — this was HYPERG(3pm) from perl v5.38.2, dated 2024-04-10. tysm for reading, go forth and compute your hypergeometric functions, you absolute main character 🎀✨