3 Mar
2014
3 Mar
'14
3:42 p.m.
H(n) = SUM(1/j, j=1..n) is the nth "harmonic number." QUESTION: what if we want to generalize this to real or complex n, not merely integer n, similarly to how Euler converted factorial to gamma function? ANSWER: Let psi(x) = d/dx ( ln Gamma(x) ) = Gamma'x) / Gamma(x) be the "psi function"; then H(x) = psi(x) + EulerMascheroniConstant = 0.5772156649 is the sought-for generalization. Euler also had found the integral representation H(x) = INTEGRAL( (1 - t^x) / (1-t), t=0..1 ) which is equivalent to this. http://en.wikipedia.org/wiki/Harmonic_number#Special_values_for_fractional_a... discusses further, and many other things are "continuizable" in similar way (also discussed in the wikipedia article).