I want the asymptotics for EllipticF(z,k) w.r.t. z (yes, not k).

My example is F:=I*EllipticF(c*I/x,k), c=2-2^(1/2),k=3+2*2^(1/2)
and I want the limit x ---> 0+.

  MultiSeries:-limit(E,x=0);

results in 0.54... + 0.54...*I (real = imag, positive value).

  MultiSeries:-series(E,x,3);

gives (after clearing the output) the same for setting x=0.


For Numerics my setting is Digits:=24 and eps:=1/10^18.

  subs(c=2-2^(1/2), k=3+2*2^(1/2), F...

I am looking for the following: given Chebychev polynomials T up
to degree n I want to express them as Legendre polynomials P.

For example one can use OrthogonalSeries[ChangeBasis] for given n.

My question is: can one give/describe/generate/... the transform
matrix for the bases directly?

The intended use is for n <= 512 (or considerably smaller), the
special case T ---> P would be enough for me.

Certainly this is classical, but I can not find it out (or looking
it up).

Does somebody have a sheet covering that algorithm for continous
anti-derivatives of rational functions (Bronstein's book §2.8
"Integration of transcendental functions")?

Or can show me, how to extract it from Maple's library in case
it is there (and I guess Maple covers that ...) for explicite
use?

What is limit( EllipticF(x*a,r), x=infinity), a and r complex?

The usual 'limit' returns unevaluated.

'MultiSeries:-limit' gives a result, but it seems to be wrong,
especially for a=I, r=1, but also for other values (no, it is
not a question of Digits):

  Tst := [alpha = (-2/(1+2*I*2^(1/2)))^(1/2), 
            rho = ((1+2*I*2^(1/2))/(1-2*I*2^(1/2)))^(1/2)];

  EllipticF(x*alpha,rho); 
  eval(%,x=2^100): # large value instead of limit 
  eval(%,Tst):
  evalf(%);

Let be q(x) some polynomial of degree = 2 in several, n variables x[i],
x to be thought as (row) vector

Can Maple provide the quadratic normalform for q (real resp. complex)?

By this it is meant that q ° f (x) equals one of

  Sum( c[i]*x[i], i=1..n)
  Sum( c[i]*x[i], i=1..n) + 1
  Sum( c[i]*x[i], i=1..n) + x[n+1]

where c[i] in K, K = Reals or Complex (should not matter so much, except
char(K), and square roots have to exist, so Rationals(squareRoots) is fine),
and f: K^n -> K^n is affine ( = bijective and linear + shift vector)?