Phrases/SpecialFunctions

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Bessel functions

Bessel functions of first kind J_n(z):

NB.*BesselJ a Bessel functions of first kind
BesselJ=: 1 : '(i.0) H. (1+m.)@(_0.25&*)@*: * ^&m.@-: % (!m.)"_'
J0=: 0 BesselJ
J1=: 1 BesselJ

This is obtained directly from representation of Bessel function as hypergeometric function.

Normal distribution

erf =: (*&(%:4p_1)%^@:*:)*[:1 H. 1.5*:    NB. A&S 7.1.21 (right)
cnd =: [:-:1:+[:erf%&(%:2)            NB. A&S 26.2.29 (solved for P)

Log Gamma

\rm{ln}\Gamma(x)=\ln(\Gamma(x))

lnG=:((%~ (%12 _360 1260 _1680)&p.@*:@%) + (-:^.2p1)+ ]-~ ^. * -&0.5)`(^.@!)@.(<&166)">

Translated directly from [Abramovitz, Stegun]

See also