Spence function

[Albert Chan]

(01-12-2021 05:41 PM)Albert Chan Wrote:  re(ln(w)*ln(1-w) - ln(-w)*ln(1+w)) = (-a)*(a/2-s*pi/2) + (a-s*pi)*(a/2) = 0

Part of the prove for I = pi^2/4 involved proving above statement.

It may be more elegantly done using geometry


arg(w) = ∠COB = θ
arg(1+w) = ∠CAB = θ/2, since |w| = 1

arg(-w) = β = θ ± pi, sign = sign(-θ)
arg(1-w) = arg(-(1+w)) = β/2

re(ln(w)*ln(1-w) - ln(-w)*ln(1+w)) = (-θ)(β/2) + (β)(θ/2) = 0