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)?

Based on some older Math group thread my problem is the following (0 < t):

F:= (x,t) -> Int(exp(-t*eta^2+x*eta)/(1+exp(eta)),eta = -infinity .. infinity);

satisfies 0 = 'diff(F(x,t),t) + diff(F(x,t),x$2)' and for that PDE Maple gives

  pdsolve(PDE, f(x,t),build): combine(%):
  subs(_c[1]=c,_C1=c1,_C2=c2,_C3=c3,%): rhs(%);
  S:=unapply(%, x,t);

    S := (x, t) -> c3*c1*exp(c^(1/2)*x-c*t)+c3*c2*exp(-c^(1/2)*x-c*t)

by separation of variables.

I am interested in t=1/2 ( to get (F(x,1/2) ) and for that define