On Mon, 12 Sept 2022 at 22:09, Fredrik Johansson <fredrik.johans...@gmail.com> wrote: > > The claim "bernoulli_plus admits a natural generalisation to real and complex > numbers but bernoulli_minus does not" (made elsewhere in this thread) seems a > bit hyperbolic. For B+ this natural generalization is -n*zeta(1-n); for B- > one can just use -n*zeta(1-n)*cos(pi*n). OK, one is a bit simpler than the > other, but both are perfectly fine entire functions.
I chose the word "natural" carefully because to me it conveys the fact that it is a subjective statement rather than a formal one. I assume it is formally possible to make infinitely many complex analytic functions to interpolate any function on the nonnegative integers (is that a named theorem?). The question is whether the factor of cos(pi*z) that you've included is something that is meaningful for noninteger values of z or something included only artifically. Did you have any reason to choose it besides fixing bernoulli(1)=-1/2 (which can also be done in infinitely many other ways)? Most mathematical functions will never gain the status of being a named function in any context so would defining a complex analytic bernoulli_minus(z) = -z*zeta(1-z)*cos(pi*z) have any particular value? Any formula involving bernoulli_plus would apply equally to bernoulli_minus but with some factors of cos(pi*z) lying around. Would any of the formulas have some other factor of cos(pi*z) for those to cancel with in order to produce something more "natural"? Would there be any value in SymPy, mpmath, Arb, Sage etc implementing such a bernoulli_minus? -- Oscar -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-devel/CAHVvXxSXzmOXcNnO3scCZOm7pZG36tGB9G-GEavqQi2hKRq-Xw%40mail.gmail.com.
