Vocabulary/hcapdot

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(m H. n) y

x (m H. n) y

Hypergeometric

Conjunction

Rank 0 0 0 -- operates on individual atoms of [x and] y, producing a result of the same shape -- WHY IS THIS IMPORTANT?

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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 =: {{((2p_0.5*y) % ^*:y) * 1 H. 1.5 *:y}}   NB. Error function
   erf 1
0.842701

   bessel1 =: {{    NB. Bessel function of the first kind Jx(y)
(((-:y)^x) % !x) * '' H. (x+1) _0.25 * *:y
}}
   1 bessel1 3
0.339059

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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 _mF_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 ₁F₁ (the confluent hypergeometric functions of the first kind) and the Bessel function a member of ₀F₁ (the confluent hypergeometric limit functions).

The most important functions for physics are those of ₂F₁, 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=: {{   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
}}

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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

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