Replying to: -------------------- |Date: Mon, 31 Jul 2000 00:13:00 -0700 (PDT) |From: "R. William Gosper" <rwg@spnet.com> |Message-Id: <200007310713.AAA23803@spnet.com> |To: math-fun@optima.CS.Arizona.EDU |Subject: atanh sum | |sum(atanh(tan(a)/tan(%pi*j/k+b)),j,1,k) = atanh(tan(a*k)/tan(b*k)); | | k | ==== | \ tan(a) tan(k a) | > atanh(--------------) = atanh(--------) . | / %pi j tan(k b) | ==== tan(----- + b) | j = 1 k | |Familiar? Expanding for small a gives the "regular polygon" sums for |the odd powers of cot, e.g., sum(cot(%pi*j/k+b),j,1,k) = k*cot(b*k); | | k | ==== | \ %pi j | > cot(----- + b) = k cot(b k) , | / k | ==== | j = 1 | |sum(cot(%pi*j/k+b)^3,j,1,k) = k*cot(b*k)*(k^2*csc(b*k)^2-1); | | k | ==== | \ 3 %pi j 2 2 | > cot (----- + b) = k cot(b k) (k csc (b k) - 1) . | / k | ==== | j = 1 ------------------------------ It may be possible to give closed forms for the sum over a period of any rationalfn(trig). A surprisingly simple result: ==== \ j pi k pi n > tan(---- + f) tan(---- + f) = - ( ) / n n 2 ==== 1<=j<k<=n A bit worse: n ==== \ j pi j pi cot(g - f) sin((g - f) n)
cot(---- + f) cot(---- + g) = n (------------------------- - 1) / n n sin(f n) sin(g n) ==== j = 1
Amazingly worse: n ==== \ pi j pi j pi j
cot(---- + f) cot(---- + g) cot(---- + h) = / n n n ==== j = 1 2 sin(h - g) cot(f n) sin(h n - g n) n (--------------------------------------- sin(g - f) sin(h - f) sin(g n) sin(h n) sin(h - f) cot(g n) sin(h n - f n)
sin(g - f) sin(h - g) sin(f n) sin(h n) sin(g - f) cot(h n) sin(g n - f n) + ---------------------------------------)/2 sin(h - f) sin(h - g) sin(f n) sin(g n) 2 2 cot(h - g) n cot(f n) sin((h - g) n) cot(h - f) n cot(g n) sin((h - f) n) + ------------------------------------- + ------------------------------------- 2 sin(g n) sin(h n) 2 sin(f n) sin(h n) 2 cos(g - f) cot(h n) cos(h - f) cot(g n) - (5 n + n + 6) (--------------------- - --------------------- sin(h - f) sin(h - g) sin(g - f) sin(h - g) cos(h - g) cot(f n) + ---------------------)/16 + (n + 3) (5 n + 2) sin(g - f) sin(h - f) cos(h - f) cot(h n) cos(h - g) cot(h n) cos(g - f) cot(g n) (--------------------- - --------------------- - --------------------- sin(g - f) sin(h - g) sin(g - f) sin(h - f) sin(h - f) sin(h - g) cos(h - g) cot(g n) cos(g - f) cot(f n) cos(h - f) cot(f n) - --------------------- - --------------------- + ---------------------)/16 sin(g - f) sin(h - f) sin(h - f) sin(h - g) sin(g - f) sin(h - g) 2 (5 n + 9 n + 6) (cot(h n) + cot(g n) + cot(f n)) + ------------------------------------------------- 8 2 cot(g - f) n sin((g - f) n) cot(h n) + ------------------------------------- 2 sin(f n) sin(g n) cos(g - f) cos(h - f) cos(h - g) - (--------------------- - --------------------- + ---------------------) sin(h - f) sin(h - g) sin(g - f) sin(h - g) sin(g - f) sin(h - f) 2 (15 n + 35 n + 18) cot(f n) cot(g n) cot(h n)/16 2 (15 n + 35 n + 18) cot(f n) cot(g n) cot(h n) + ----------------------------------------------, 8 from a harrowing derivation via 16-page intermediate results, including a quadrivariate generating fcn to circumvent a *nasty* cubic. This undermines my confidence in being able to automate such sums (and products). Here's the stringout if you want to test or simplify it: 'SUM(COT(%PI*J/N+F)*COT(%PI*J/N+G)*COT(%PI*J/N+H),J,1,N) = N^2*(SIN(H-G)*COT(F*N)*SIN(H*N-G*N)/(SIN(G-F)*SIN(H-F)*SIN(G*N)*SIN(H*N))-SIN(H-F)*COT(G*N)*SIN(H*N-F*N)/(SIN(G-F)*SIN(H-G)*SIN(F*N)*SIN(H*N))+SIN(G-F)*COT(H*N)*SIN(G*N-F*N)/(SIN(H-F)*SIN(H-G)*SIN(F*N)*SIN(G*N)))/2+COT(H-G)*N^2*COT(F*N)*SIN((H-G)*N)/(2*SIN(G*N)*SIN(H*N))+COT(H-F)*N^2*COT(G*N)*SIN((H-F)*N)/(2*SIN(F*N)*SIN(H*N))-(5*N^2+N+6)*(COS(G-F)*COT(H*N)/(SIN(H-F)*SIN(H-G))-COS(H-F)*COT(G*N)/(SIN(G-F)*SIN(H-G))+COS(H-G)*COT(F*N)/(SIN(G-F)*SIN(H-F)))/16+(N+3)*(5*N+2)*(COS(H-F)*COT(H*N)/(SIN(G-F)*SIN(H-G))-COS(H-G)*COT(H*N)/(SIN(G-F)*SIN(H-F))-COS(G-F)*COT(G*N)/(SIN(H-F)*SIN(H-G))-COS(H-G)*COT(G*N)/(SIN(G-F)*SIN(H-F))-COS(G-F)*COT(F*N)/(SIN(H-F)*SIN(H-G))+COS(H-F)*COT(F*N)/(SIN(G-F)*SIN(H-G)))/16+(5*N^2+9*N+6)*(COT(H*N)+COT(G*N)+COT(F*N))/8+COT(G-F)*N^2*SIN((G-F)*N)*COT(H*N)/(2*SIN(F*N)*SIN(G*N))-(COS(G-F)/(SIN(H-F)*SIN(H-G))-COS(H-F)/(SIN(G-F)*SIN(H-G))+COS(H-G)/(SIN(G-F)*SIN(H-F)))*(15*N^2+35*N+18)*COT(F*N)*COT(G*N)*COT(H*N)/16+(15*N^2+35*N+18)*COT(F*N)*COT(G*N)*COT(H*N )/8; --rwg PS, given that my Macsyma was last compiled for a 286, Mma's far greater capacity was a temptation, except that 5.2, at least, still lacks the equivalent of POWERSERIES for extracting the general term of (a certain derivative at a certain point of) that generating function. PPS, my former student aced his tests but skipped his homework, and is now repeating Algebra I. As soon as I can photocopy a few pages, I'll share some of the stupendous howlers I found on a quick perusal of his new text. Amazingly, it's *not* Prentice-Hall, so it must be that the "educators" actively seek texts by and for morons. --------------------------------- Luggage? GPS? Comic books? Check out fitting gifts for grads at Yahoo! Search.