Turns out I still had typos. Trying again. Goal is to devise a function Bernoulli(n) defined for complex n (not nec'ly integers) and such that reals map to reals, and agrees with Bernulli numbers when n=0,1,2,3,... http://en.wikipedia.org/wiki/Bernoulli_number My original suggestion with typos hopefully now finally corrected, using a ln(z) definition slit along the positive real axis (which is not the standard definition), and A^B = exp(B*ln(A)), was Bernoulli(n) = Gamma(n+1)/(2*pi*i) * integral z^(-n) / [exp(z)-1] dz using an anticlockwise contour of integration like x = y^2 - 1. However, saying that about the nonstandard ln was somewhat cavalier of me. If we use the standard ln(z) definition with slit along the negative real axis, then Bernoulli(n) = Gamma(n+1)/(4*pi*i) * integral [ exp(n*[pi*i-ln(-z)]) + exp(n*[-pi*i-ln(-z)]) ] / [exp(z)-1] dz would work where the average of the two exp's (and the reflection symmetry of the contour) serves to cancel out imaginary parts so that Bernoulli(n) is real if n is real. However, Asimov may wish to reconsider his desire that it map real-->real. To explain, consider the functions sin(x) and cos(x), which both map real-->real. Do you not think exp(i*x) is somehow more fundamental? If so you might prefer something like Bernoulli(n) = Gamma(n+1)/(2*pi*i) * integral exp(n*[pi*i-ln(-z)]) / [exp(z)-1] dz which does not map reals to reals, but its real part is my previous definition...