An Equation of Euler

Our starting point is a simple definite integral and the value of a hypergeometric sum.

f := t -> (1-t)/((1+t)*sqrt(t)):
Int(f(t),t=0..1) = int(f(t),t=0..1);
evalf(Pi-2,32);

hypergeom([1,2],[5/2],1/2)/(3/2):
simplify(%); evalf(%,32);

Note that Maple knows the exact value of the integral, but does not give a closed expression for the hypergeometric sum, so probably is not aware of the following identity Leonhard Euler already knew.

print("***");
Int((1-t)/((t+1)*sqrt(t)),t = 0 .. 1) = 2/3*hypergeom([1, 2],[5/2],1/2);

In fact, this is only a special case of an integral representation of the Gauss hypergeometric function which Euler discovered. How can we prove this equation? One way is to show, that both sides satisfy the same holonomic differential equation and initial values. This approach is explained in Wolfram Koepf, 'Hypergeometric Summation' (ref. [1]), page 201. example 12.4. There you can also find some state of the art algorithms implemented for Maple. If you want to stay at the leading edge of symbolic math technology, get this book and the software, highly recommended! The procedures 'sumdiffeq', 'hyperterm' and 'intdiffeq' can be found there.

a := 1: b := 2: c := 5/2:
sumdiffeq(hyperterm([a,b],[c],x,k),k,S(x));

intdiffeq(GAMMA(c)/(GAMMA(b)*GAMMA(c-b))*
t^(b-1)*(1-t)^(c-b-1)*(1-t*x)^(-a),t,S(x));

However, we are here not interested in the general Euler integral representation itself, rather in the value it evaluates to in our special case, .
We will take formula (***) as the center of our investigation.
This worksheet records my exploration of this equation in hindsight of using it in the numerical computation of .

This note was inspired by the reading of "Pi - die Story" by Jean-Paul Delahaye, published by Birkhauser Verlag 1999, a translation of "Le fascinant nombre Pi", published by Pour la Science in 1997. I highly recommend this book (see ref. [2]).