[VA] SRC #010 - Pi Day 2022 Special

[Albert Chan]

(03-16-2022 08:49 PM)Albert Chan Wrote:  \(\displaystyle \ln(\pi) = \int_0^{1/2} \left(
\frac{1}{y} + \frac{\pi (2y-1)}{\tan(\pi y)}
\right) dy\)

Wolfram Alpha proved this, with closed-form anti-derivative !

Code:

def G(y):
    k = pi*j
    z = exp(2*k*y)
    return ln(y) - ln(sin(pi*y)) + y*(2*log1p(-z)-k*y) + polylog(2,z)/k

>>> from mpmath import *
>>> limit(G,1/2) - limit(G,0)
mpc(real='1.1447298858494002', imag='0.0')
>>> exp(_)
mpc(real='3.1415926535897931', imag='0.0')

OP product form, which integral was derived from, is thus proved.