From: Mike Stay <metaweta@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Wednesday, July 1, 2009 3:54:09 PM Subject: Re: [math-fun] Addition formula paper Thank you, Simon, this is an interesting paper! However, it's unfortunately not the one I was looking for; also, I remember better the problem: given a function (+) of two variables x,y, find a function p() such that p(x) + p(y) = p(x (+) y) Somehow they got from x (+) y = xy to p'(x) = 1/x and then integrated to get p = log. -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com _______________________________________________ Assume that p is differentiable. For the particular case, p(xy) = p(x) + p(y), we have p(1x) = p(x) + p(1), so that p(1) = 0. p(x (1 + dy)) = p(x) + p(1 + dy) p(x) + p'(x) x dy = p(x) + p(1) + p'(1) dy p'(x) = p'(1) / x -- Gene

My collection for function s site includes all formulas for all arctrigs with structure a ArcTrig[z]+b OtherArctrig[z]== c ArcTrig4 +Pi d. They are correct for all values of arguments. We have corresponding program for such operations. But it was not used yet in Mathematica deeply and was not shown to public in Internet yet. For example: In[20]:= Chop[Table[ a*ArcCsc[x] + b*ArcTanh[y] - (-2*I*Pi*(Floor[(-Arg[1/((Sqrt[1 - 1/x^2] + I/x)^(I*a)*(1 - y)^(b/2))] - Arg[(y + 1)^(b/2)] + Pi)/(2*Pi)] + Floor[(Pi - Im[Log[1/((Sqrt[1 - 1/x^2] + I/x)^(I*a)*(1 - y)^(b/2))]])/ (2*Pi)] + Floor[(Pi - (1/2)*Im[ b*Log[y + 1]])/(2*Pi)]) - 2*I*Pi*(Floor[(-Arg[(Sqrt[1 - 1/x^2] + I/x)^ ((-I)*a)] - Arg[(1 - y)^(-(b/2))] + Pi)/(2*Pi)] + Floor[ ((1/2)*Im[b*Log[1 - y]] + Pi)/(2*Pi)] + Floor[(Re[a*Log[Sqrt[1 - 1/x^2] + I/x]] + Pi)/(2*Pi)]) + I*Pi* (1 - (-1)^(Floor[-(Arg[(y + 1)^(b/2)/ ((1 - y)^(b/ 2)*(Sqrt[1 - 1/x^2] + I/x)^(I*a)) + 1]/(2*Pi))] - Floor[-(Arg[(y + 1)^(b/2)/ ((Sqrt[1 - 1/x^2] + I/x)^(I*a)* (1 - y)^(b/2))]/(2*Pi))])) + I*(-1)^(Floor[-(Arg[(y + 1)^(b/2)/ ((Sqrt[1 - 1/x^2] + I/x)^(I*a)* (1 - y)^(b/2))]/Pi)] + Floor[Arg[(y + 1)^(b/2)/((Sqrt[1 - 1/x^2] + I/x)^(I*a)*(1 - y)^(b/ 2)) - 1]/(2*Pi) - Arg[(y + 1)^(b/2)/ ((1 - y)^(b/2)*(Sqrt[1 - 1/x^2] + I/x)^(I*a)) + 1]/(2*Pi) + 1/2])* (ArcCsc[(2*(y + 1)^(b/2))/ ((Sqrt[1 - 1/x^2] + I/x)^(I*a)* (1 - y)^(b/2)*((y + 1)^b/((1 - y)^b* (Sqrt[1 - 1/x^2] + I/x)^(2*I* a)) + 1))] - Pi/2)) /. {a -> Random[Complex]} /. {b -> Random[Complex]} /. {x -> (j + Random[])*Exp[Pi*I*(k/4)]} /. {y -> (i + Random[])*Exp[Pi*I*(n/4)]}, {k, 0, 7}, {j, 0, 1}, {n, 0, 7}, {i, 0, 1}]] Out[20]= {{{{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}, {{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}}, {{{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}, {{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}}, {{{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}, {{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}}, {{{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}, {{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}}, {{{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}, {{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}}, {{{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}, {{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}}, {{{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}, {{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}}, {{{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}, {{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}}} Oleg rwg@sdf.lonestar.org wrote:

...

I'm looking for a paper I read once (more than six years ago) on addition formulas. It had the tangent addition formula as a special case: atan(x) + atan(y) = atan((x+y)/(1-xy)) as well as log log(x) + log(y) = log(xy) and various other ones. The idea was to find a function p and an operator @ such that p(x) + p(y) = p(x@y), and they showed how to come up with a large class of these. It also, as I recall, had something to do with intersecting chords of a conic.

Does anyone know what paper this was, or could anyone suggest keywords to google for it? -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com

This ain't it, but, due to its non-announcement at the time (ca 2001), some of you may have missed that bilinear (Somos) sequences have addition formulae: http://arxiv.org/PS_cache/math/pdf/0703/0703470v1.pdf --rwg