Evaluating the Dilogarithm Function

Among the many fascinating mathematical curiosities is the so-called dilogarithmic function. It was believed to be originally discussed by Leonard Euler in 1729 in the form:

Li₂ (x) =  å _n=1   [xⁿ  / n²]                            

Where the summation is to infinity (oo) and Euler was unable to evaluate Li₂ (1)   from the integral form. However, Landen, in a paper appearing in Philosophical Transactions (Landen, John: 1760,  ‘A New Method of Computing The Sums of Certain Series, Vol. 51, Pt. 2, p. 553)showed the evaluation required the use of log (-1). Unaware of Euler’s work, Landen set out to determine log (-1) by first differentiating x =   sin (z) to obtain: dz/ i=   dx/ [x² -1]^1/2

Integrating this, he got:

z/ i = log (x + [x² -1]^1/2  / i


where i= [-1]^1/2

Landen then set z = π/2 and x = 1 so that log ([-1]^1/2) = log(i) = -π/2i

Since a square root must have two values, he concluded: log (-1) = +/-  π/ i

Then to evaluate Li₂ (1) he commenced with the series form:

x^-1 + x^-2 /2 + x ^-3/ 3 + ……= log (1/ 1 – x) = log x + log (1/ 1- x) - π/i


Landen then divided the preceding by x to obtain:


- Li₂ (1/x) = - π/i log x + ½ (log x)² + Li₂ (x) + C


To find C, Landen took x = 1, and then using algebra (solving for C above) obtained:


C = -2 å ^¥ _n=1  (1/n²)


He then had to evaluate the series above and did this by setting x = -1 in the expression for - Li₂ (1/x) . Using a (+) sign this time, he arrived at:


å ^¥ _n=1  (1/n²) = π² / 6


Landen’s computations indicated that the formula for - Li₂ (1/x) was only correct for the case x > 1. His tactic of dividing by x, then integrating led ultimately to the recognition that the series Li₂ (x) and Li₂ (1/x) converged when x = exp(i Θ) for the interval:

0 < Θ < 2π


Note, however, that if this is substituted directly into the expression for - Li₂ (1/x) the result is invalid. However, choosing the (+) sign instead makes it valid, yielding the Fourier series expansion of the 2nd Bernouli polynomial.