Asymptotic series have been extended to allow exponential terms to appear in the resulting series. Example:
> asympt( Psi(2*exp(x))-x, x, 4 );
1 1 1
ln(2) - -------- - ---------- + O(-------)
4 exp(x) 2 4
48 exp(x) exp(x)
> limit( Psi(2*exp(x))-x, x=infinity );
ln(2)
Series expansions for erfc(x), GAMMA(a,x) for
and MeijerG(a,b,x) for
,
have been added.
Also asymptotic expansions
for GAMMA(x) and binomial(n,k). Examples:
> asympt( GAMMA(x)*Ei(x)/x^x/sqrt(2*Pi), x );
1 13 601 319721 60973877 1
---- + ------- + -------- + ---------- + ------------- + O(-----)
3/2 5/2 7/2 9/2 11/2 13/2
x 12 x 288 x 51840 x 2488320 x x
# An indefinite summation using Gosper's algorithm
> s := sum( binomial(2*n,n)/(n+1)/(2^n)^2, n );
binomial(2 n, n)
s := - 1/2 ----------------
(n - 1) 2
(2 )
> asympt(s,n,4);
2 1 1 1
- ---------- + ------------ - ------------- + O(----)
1/2 1/2 1/2 3/2 1/2 5/2 7/2
Pi n 4 Pi n 64 Pi n n
# Hence since s = -2 at n = 0, we have
> sum( binomial(2*n,n)/(n+1)/(2^n)^2, n=0..infinity );
2