sage: var('z');
sage: f(z) = 1 / (1+exp(z)) / (z^2+pi^2)
sage: f(z).residue( z==i*pi )
-1/4*I/pi - 1/4/pi^2
sage: bool( f(z).residue( z==i*pi ) == 1/(2*pi*i) * (1/2 + 1/(2*pi*i)) )
True

sage: f.residue( z==3*pi*i )
z |--> 1/8/pi^2
sage: f.residue( z==5*pi*i )
z |--> 1/24/pi^2
sage: f.residue( z==7*pi*i )
z |--> 1/48/pi^2
sage: f.residue( z==9*pi*i )
z |--> 1/80/pi^2
sage: f.residue( z==11*pi*i )
z |--> 1/120/pi^2

? p 1000
   realprecision = 1001 significant digits (1000 digits displayed)
? intnum( x=-200, +100, 1 / (1+exp(x)) / (x^2+Pi^2) )
%19 = 0.49500041117264675924354715963413516...