m,p = var('m,p')
#taylor coefficient for erf(3x)
a_erf(m) = (3)^(2*m+1)*(-1)^m*2/sqrt(pi)/(factorial(m)*(2*m+1))
#coefficient of chebyshev polynomial
c_erf_cheb(p) = sum(a_erf(m)*binomial(2*m+1,m-p)*4^-m,m,p,oo).simplify_full()
-6/11*(bessel_I(6, -9/2) - bessel_I(5, -9/2))*sqrt(e)*e^(-5)/sqrt(pi)
which, to me, is a very useful answer. But other sums are simply wrong.
k = var('k')
sum(x^(2*k)/factorial(2*k),k,0,oo)
gives
-1/4*sqrt(2)*sqrt(pi)*x^(3/2)
but the answer should besinh(x)
. For other sums, Sage simply repeats what I told it.
sum(x^(3*k)/factorial(2*k),k,0,oo)
I understand that Sage has limited exploitation of Maxima's hypergeometric functionality, and I suspect this is the main issue. Are there any conceivable workarounds?
But other sums are simply wrong.
k = var('k')
sum(x^(2*k)/factorial(2*k),k,0,oo)
gives
-1/4*sqrt(2)*sqrt(pi)*x^(3/2)
but the answer should besinh(x)
.
For other sums, Sage simply repeats what I told it.
sum(x^(3*k)/factorial(2*k),k,0,oo)
I understand that Sage has limited exploitation of Maxima's hypergeometric functionality, and I suspect this is the main issue. Are there any conceivable workarounds?
I understand that Sage has limited exploitation of Maxima's hypergeometric functionality, and I suspect this is the main issue. Are there any conceivable workarounds?
> which, to me, is a very useful answer. But other sums are simply wrong.
>
> k = var('k')
> sum(x^(2*k)/factorial(2*k),k,0,oo)
I'm working with Maxima 5.33.0. I get
simplify_sum ('sum(x^(2*k)/factorial(2*k),k,0,inf));
=> sqrt(%pi)*bessel_i(-1/2,x)*sqrt(x)/sqrt(2)
which seems to be cosh(x).
> sum(x^(3*k)/factorial(2*k),k,0,oo)
I get
simplify_sum ('sum(x^(3*k)/factorial(2*k),k,0,inf));
=> sqrt(%pi)*bessel_i(-1/2,x^(3/2))*x^(3/4)/sqrt(2)
sum(x^(3*k)/factorial(2*k),k,0,oo)
I understand that Sage has limited exploitation of Maxima's hypergeometric functionality, and I suspect this is the main issue. Are there any conceivable workarounds?