Thanks RV...

(Nice forum you guys and gals have here, btw

)

And you're quite right about the interchangeability - for want of a better word - between values of the Clausen function and polygamma function.

Incidentally, if you fancy another transcendental approach, the Inverse Tangent Integral is also expressible in terms of the Barnes' G-Function (reciprocal of the double gamma function - not to be confused with the digamma function).

Define multiple gamma function as follows:

\(\displaystyle \Gamma_n(1) = 1\)

\(\displaystyle \Gamma_1(z) = \Gamma(z)\)

\(\displaystyle \Gamma_{n+1}(z+1)=\frac{\Gamma_{n+1}(z)}{\Gamma_n(z)}\); for \(\displaystyle n \in \mathbb{Z}\) and \(\displaystyle z \in \mathbb{C}\)

If the condition of convexity is added, then

\(\displaystyle (-1)^{n+1}\frac{d^{n+1}}{dz^{n+1}}\log\Gamma_n(z) \ge 0,\,\,\, n > 0\)

has a unique solution.

Then the Barnes' G-Function - \(\displaystyle \text{G}(z)\) - satisfies

\(\displaystyle \text{G}(1)=\text{G}(2)=\text{G}(3)=1\)

\(\displaystyle \text{G}(1+z)=\Gamma(z)\text{G}(z)\)

\(\displaystyle \frac{d^3}{dz^3}\text{G}(z)\ge 0,\,\,\, z>0\)

Furthermore, the Barnes' G-function can be expressed in terms of the Clausen function - and vice versa - via the identity

\(\displaystyle \log\left(\frac{\text{G}(1+z)}{\text{G}(1-z)}\right)=-z\log\left(\frac{\sin \pi z}{\pi}\right)-\frac{1}{2\pi}\text{Cl}_2(2\pi z)\)

for 0 < z < 1. And so, within certain limits

\(\displaystyle \text{Ti}_2(\tan\tfrac{p\pi}{q})=\pi\log\left[\frac{\text{G}(1-\tfrac{p}{q})\text{G}(\tfrac{1}{2}-\tfrac{p}{q})}{\text{G}(1+\tfrac{p}{q})\text{G}(\tfrac{3}{2}+\tfrac{p}{q})}\right]-\frac{\pi}{2}\log\left(\cos\frac{p\pi}{q}\right)+\frac{\pi(2p+q)}{2q}\log\pi\)

I've not seen this result in any books or papers, but it's easy enough to deduce, so I can't imagine it's new....

Incidentally, going back to what you said about the polygamma function, it's worth noting the following functional relation for the the Barne's G-function (for Re(z) > 0):

\(\displaystyle \log\text{G}(z+1)-z\log\Gamma(z) = \zeta'(-1)-\zeta(-1, z)\)

In a very loose sense, this identity, in conjunction with previous results above, makes the connection between the Inverse Tangent Integral and derivatives of the Hurwitz Zeta function, Barnes G-function, Clause function, loggamma function - and much more besides - abundantly clear (it's an easy step to wander into the zone of Dirichlet Beta functions, Zeta functions, Polylogarithms, Lerch Transcendents, et al).

If you fancy a good read about the Barne's G-function, I'd highly recommend "Contributions to the theory of the Barnes' Function", by Victor S. Adamchik. It's widely available on-line in PDF form (

http://repository.cmu.edu/cgi/viewconte ... xt=compsci)

And if that link doesn't work, just Google "Contributions to the theory of the Barnes' Function"...

Gethin