Okay, I know the answer now. But I dare anyone to discover it from the existing J Help documentation of (H.) !
Including Concrete Math Companion, Vector, and Ewart Shaw's posting !! Your mission, should you accept it, is to define a verb F that accepts an argument in (roughly) the syntax of Abramowitz and Stegun (A&S) chapter 15, viz F(a;b;c;z), and calls (H.) with the correct arguments. Here's some examples drawn from A&S... ln=: ^. arcsin=: _1&o. arctan=: _3&o. ] z=: 5%~ i.6 0 0.2 0.4 0.6 0.8 1 F(1;1;2;z) NB. [15.1.3] -(ln 1-z)%z 0 1.11572 1.27706 1.52715 2.0118 _ F(1r2;1;3r2;z^2) NB. [15.1.4] -:(ln (1+z)%(1-z))%z 0 1.01366 1.05912 1.15525 1.37327 _ F(1r2;1;3r2;-z^2) NB. [15.1.5] (arctan z) %z 0 0.986978 0.951266 0.900699 0.843426 0.785398 F(1r2;1r2;3r2;z^2) NB. [15.1.6] (arcsin z) %z 0 1.00679 1.02879 1.0725 1.15912 1.5708 It's 4 instances of the Hypergeometric Series (F) with the functions it is supposed to approximate when 0<(|z)<1. (So for z=0 and z=1 the results can't be expected to match. But I've included these values in z anyway.) Hint: call H. with left argument x=50 (the number of terms of the series to be summed) as it can take a long time if you let it go to the limit by calling it monadically. Just to preempt someone splitting hairs, no my J syntax of A&S's F(a,b;c;z) isn't quite the same. A&S has a comma as the first separator, whereas I've a semicolon. IanClark On Sat, Jan 18, 2014 at 6:09 PM, Mike Day <[email protected]>wrote: > Ewart Shaw wrote about these, so look for his emails on the subject > failing other channels. He might like to comment for himself, of course, > if his e-address (as I have it, above) is still correct. > > Mike > > > On 18/01/2014 11:01, Ian Clark wrote: > >> Just one empty stub remains in the Accessible Dictionary (aka NuVoc >> --remember it?): >> >> H. (Hypergeometric) Conjunction >> http://www.jsoftware.com/jwiki/Vocabulary/hcapdot >> >> Once that's filled-in, then NuVoc is more-or-less ready to go. You can >> already see it at http://www.jsoftware.com/jwiki/Vocabulary >> But alas, I need help... >> >> The J Dictionary (the old one) references Abramowitz and Stegun (A&S), >> Chapter 15: Hypergeometric Functions. Now A&S represent the syntax of the >> general case like so: >> >> F(a; b; c; z) >> >> Both NuVoc and the J Dictionary present the syntax of the (H.) primitive >> like so: >> >> (m H. n) y >> >> where both m and n are numeric lists. >> >> Now suppose I'm a newbie, and my first sight of: >> http://www.jsoftware.com/help/dictionary/dhcapdot.htm >> just gives me a dull ache between the eyes. I need clear, unambiguous >> instructions for taking any example I choose from A&S and mapping it onto: >> (m H. n)y >> >> Let me make a start: >> z --> y >> That was the easy bit. Now... how should (a; b; c) --> (m; n)? >> >> Or should I be asking: how *best* should (a; b; c) be mapped onto m and n? >> Because as I see it, it's ambiguous. Just for starters: >> >> F(a; b; c; z) = F(b; a; c; z) -----[A&S 15.1.1] >> >> Suggestions please. >> >> IanClark >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm >> >> > > --- > This email is free from viruses and malware because avast! Antivirus > protection is active. > http://www.avast.com > > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
