Vocabulary/hcapdot

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

Vocabulary/rank000


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:

Hyp.jpg

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

Asx.jpg

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