# Vocabulary/hcapdot

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`(m H. n) y``x (m H. n) y`Hypergeometric Conjunction

`x (m H. n) y ` sums `x` terms of a *generalized hypergeometric series*.

- Operands
`m`and`n`describe the series - Argument
`y`gives the argument values(s) (may be real or complex).

Omitting `x` gives the limiting case as `x` tends to infinity.

erf =: 3 : '(((2p_0.5)*y) % (^*:y)) * 1 H. 1.5 *: y' NB. Error function erf 1 0.842701 bessel1 =: dyad define NB. Bessel function of the first kind Jx(y) (((-: y)^x) % !x) * '' H. (x+1) _0.25 * *: y ) 1 bessel1 3 0.339059

### Common Uses

Many common mathematical functions can be computed as *instantiations* of the generalized hypergeometric series
by a choice of the operands `m` and `n` (atoms or lists).

These are classified into families, designated *,,m,,F,,n,,* according to the numbers of values in `m` and `n`.
Each family has many members depending on the actual values of `m` and `n`.

The error function above is a member of *,,1,,F,,1,,* (the *confluent hypergeometric functions of the first kind*)
and the Bessel function a member of *,,0,,F,,1,,* (the *confluent hypergeometric limit functions*).

The most important functions for physics are those of *,,2,,F,,1,,*, which are called the *hypergeometric functions*.

Chapter 15 of Abramowitz & Stegun represent any given instance of the hypergeometric functions
by *F(a,b;c;z)*, where *a*, *b* and *c* are constants and *z* is a point in the object domain, the complex plane:

The notation *(a),,n,,* is the *rising Pochhammer symbol*, implemented in J as the stope function ` (a ^!.1 n)`

To convert this *F*-notation to J syntax: ` (m H. n) y`

m=: a,b n=: c y=: z

A convenient verb for this purpose is

F=: 3 : 0 NB. Convert F(a;b;c;z) into monadic H. call 'a b c'=. 3{.y z=. > 3}.y m=. a,b n=. c m H. n z )

### Examples

Abramowitz & Stegun, Chapter 15

Ancillary verbs for sample functions

ln=: ^. arcsin=: _1&o. arctan=: _3&o.

Sample points in the object domain (the disk of convergence 1>|z)

] z=: }. 5%~ i.5 0.2 0.4 0.6 0.8

Identities 15.1.3 to 15.1.6 with their equivalent functions

F(1; 1; 2; z) NB. 15.1.3 1.11572 1.27706 1.52715 2.0118 -(ln 1-z) % z 1.11572 1.27706 1.52715 2.0118 F(1r2; 1; 3r2; z^2) NB. 15.1.4 1.01366 1.05912 1.15525 1.37327 -:(ln (1+z)%(1-z)) % z 1.01366 1.05912 1.15525 1.37327 F(1r2; 1; 3r2; -z^2) NB. 15.1.5 0.986978 0.951266 0.900699 0.843426 (arctan z) % z 0.986978 0.951266 0.900699 0.843426 F(1r2; 1r2; 3r2; z^2) NB. 15.1.6 1.00679 1.02879 1.0725 1.15912 (arcsin z) % z 1.00679 1.02879 1.0725 1.15912