Analogues of the Gamma Function

The first 14 sections of Chapter 8 comprise but 41/2 of the 12 pages in this chapter. Initial results are concerned with partial sums of the harmonic series and the logarithmic derivative ψ(x) of the gamma function. As might be expected, most of these results are very familiar. Ramanujan actually does not express his formulas in terms of ψ(x) but instead in terms of $$\varphi \left( x \right) = \sum\nolimits_{k = 1}^x {1/k.} $$. As in Chapter 6, Ramanujan really intends ϕ(x) to be interpreted as ϕ(x + 1) + γ, for all real x, where γ denotes Euler’s constant. These 14 sections also contain several evaluations of elementary integrals of rational functions. Certain of these integrals are connected with an interesting series $$\sum\nolimits_{k = 1}^\infty {1/\left\{ {{{\left( {kx} \right)}^3} - kx} \right\},} $$, which Ramanujan also examined in Chapter 2.