Can we used the Reimann sum to get the result??

This is the code:

I know that the taylor function of a cosine is a series, then  I can write this:

Int(cos((x-1)*h)/x^(5/3), x = 1 .. infinity);

IntegrationTools:-Change(%, x = xi+1);

sum((-1)^n*h^(2*n)*(Int(xi^(2*n)/(xi+1)^(5/3), xi = 0 .. infinity))/factorial(2*n), n = 0 .. infinity);

value(%)

Yes I have this result, but try to do this differetly. You use the change of variable as : q = x-1 after that you use taylor command to get the cos develpment series. You replace in the integral and you will do another change of variable : q+1 =1/t to get the beta function. At the end resum with maple and you will find the same result but at the place of Lommels2 you get LommelS1.

Very Weird.

Excuse me but It begins at 1 instead of zero.

Yes exaclty. forgive me