49 years ago, I used that very (math, not Mma) fact to prove (1/Gamma(s)) Integral_{0,1} ((1-x^r)/(1-x)) ln(1/x)^(s-1) dx = 1 + 1/2^s + . . . + 1/r^s for r a positive integer (and thus a way to interpolate between these partial Zeta sums, by letting r be a non-integer). In the limit this becomes Gamma(s) Zeta(s) = Integral_{0,1} (ln(1/x)^(s-1) / (1-x)) dx which I thought was really cool, until entering college in September and telling this to my freshman advisor (Henry McKean), upon which he reached over and picked up a Whittaker and Watson on his shelf, flipped through it, and showed me essentially the identical formula there. (I was devastated.) --Dan On 2013-08-12, at 11:07 PM, Bill Gosper wrote:
Am I the only one who hasn't seen
In[650]:= FullSimplify[Integrate[Log[1/z]^n, {z, 0, 1}], n \[Element] Integers && n > 0]
Out[650]= n!
until today?