Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
1995
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 1995

[Date Index] [Thread Index] [Author Index]

Search the Archive

Re: ArcSin

  • To: mathgroup at christensen.cybernetics.net
  • Subject: [mg1570] Re: ArcSin
  • From: bob Hanlon <hanlon at pafosu2.hq.af.mil>
  • Date: Sun, 2 Jul 1995 00:02:33 -0400

Needs["Algebra`SymbolicSum`"]

Expanding the ArcSin about x=0

Series[ArcSin[x], {x, 0, 13}]

     3      5      7       9       11        13
    x    3 x    5 x    35 x    63 x     231 x         14
x + -- + ---- + ---- + ----- + ------ + ------- + O[x]
    6     40    112    1152     2816     13312

See Abramowitz and Stegun, 4.4.40; Hansen, 5.22.14

SymbolicSum[Pochhammer[1/2, n]/(n! (2n+1)) x^(2n+1), 
	{n, 0, Infinity}]

ArcSin[x]

An equivalent expression without the Pochhammer symbol

SymbolicSum[2 (2n)!/((n!)^2 (2n+1)) (x/2)^(2n+1), 
	{n, 0, Infinity}]

ArcSin[x]

If by a recursive relation you mean how to calculate the n-th term of the 
series from the (n-1) term, the ratio of the terms is given by

r = ( 2 (2n)!/((n!)^2 (2n+1)) (x/2)^(2n+1) ) / 
	( ( 2 (2n)!/((n!)^2 (2n+1)) (x/2)^(2n+1) ) /. 
	n -> n-1 ) // Simplify

            2          2
(-1 + 2 n) x  (-1 + n)!  (2 n)!
-------------------------------
                             2
 4 (1 + 2 n) (2 (-1 + n))! n!

r = r /. { n! -> n (n-1)!, (2n)! -> (2n)(2n-1) (2(n-1))! }

          2  2
(-1 + 2 n)  x
--------------
2 n (1 + 2 n)

t[0, x_] := x
t[n_Integer?Positive, x_] := t[n, x] = 
	(2n-1)^2 x^2 / (2n (2n+1)) t[n-1, x]

Table[t[n, x], {n, 0, 6}]

     3     5     7      9      11       13
    x   3 x   5 x   35 x   63 x    231 x
{x, --, ----, ----, -----, ------, -------}
    6    40   112   1152    2816    13312


>  From: njachimi at glibm10.cen.uiuc.edu (Nathan Jachimiec)
>  Newsgroups: comp.soft-sys.math.mathematica
>  Subject: arcsin
>  Date: 23 Jun 1995 02:39:29 GMT
>  
>  Is there a recurrence relation that one can use to compute an arcsin???
>  How can one numerically compute the value of arcsin with a recurrence 
>  relation (restated for clarity)?  Can someone direct me to some
resource 
>  or formula for numerical solutions of this function...
>  
>  njachimi at uiuc.edu





  • Prev by Date: rhs of SetDelayed
  • Next by Date: Bug in Mma 2.2.2 for Windows?
  • Previous by thread: Re: arcsin
  • Next by thread: Mathematica Assignments