On Tue, Sep 13, 2022 at 10:35 AM Oscar Benjamin <oscar.j.benja...@gmail.com> wrote: > > 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?).
you can always add sin(x)/pi - which vanishes on 0,1,2,..., - to your interpolant. So that's an excercise :-) > > 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. -- 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/CAAWYfq1ShHLMK2VBSJGpsHoU7YsZ-sfU9h%2BaUboLoHXFkjETQg%40mail.gmail.com.
