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 mFn 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=: 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