Downrate my excitement over the Abel-Plana formula, inf inf inf ==== / / \ [ h(%i y) - h(- %i y) [ h(0) > h(n) = %i I ------------------- dy + I h(x) dx + ----, / ] 2 %pi y ] 2 ==== / %e - 1 / n = 0 0 0 which is like Euler-Maclaurin but gives exact identities instead of divergent approximations. Reason: it can be wrong! E.g., for h(n):=q^n^2*e^(a*n), inf inf inf ==== 2 / / 2 \ a n n [ sin(a y) [ x a x 1
%e q = - 2 I ------------------- dy + I q %e dx + - / ] 2 ] 2 ==== / y 2 %pi y / n = 0 0 q (%e - 1) 0
is only meaningful for q on the unit circle, (e^(i b)). (Is Limbaugh still on?). But even though everything then converges, it's wrong! According to http://dlmf.nist.gov/2/10/ Sufficient conditions for the validity of this [...] result are: 1.(a) On the strip 0<=Re(z)<oo, h(z) is analytic in its interior, <the derivatives of h are> continuous on its closure, and h(z)=o(exp(2 pi Im(z)) as Im(z)->±oo, uniformly with respect to 0<=Re(z)<oo. (b) h(z) is real when 0<=z<oo. Geez, Edith, I think those shysters voided our warranty. --rwg PS, rtw>Some detail regarding <paleoballistics>:
The telegram with Gosper's P30 gun coordinates was dated 4 Nov70.
Gosper's puffer train and P46 gun discovery reported Aug/Sep 71.
Cordership reported Dec 71. [Actually, c/12 puffer]
Schick's puffers reported early (Feb?) 72. R. Wainwright
Also, correction: guns of all true periods >=60 are known, *except* 61. INTERSPECIES REST IN PIECES