Dear Bill: I assume that some of the same types of formulae work for hyperbolic sines? I'm still looking for good formulae for the following cases: sinh(x*2^k) = ?? sinh(x+y) = ?? asinh(x*2^k) = ?? asinh(x+y) = ?? Thx. Henry Baker hbaker1@pipeline.com Santa Barbara, CA At 02:32 AM 10/24/2007, you wrote:
n n ==== /===\ \ k | | %pi k > (- 1) | | 2 sin(a - -----) n / | | j n + 1 ==== ==== j = 0 \ k = 0 sin( > a ) = ------------------------------------, / k 2 (n + 1) ==== k = 1 where %pi n a := -----. 0 2
There are n^2 sins on the right, vs n2^(n-1) for the usual addition formula. But all n^2 sins are different, vs only 2n species for the usual formula. E.g., for n=3,
%pi %pi sin(a + a + a ) = - sqrt(2) sin(a + ---) sin(a + ---) 3 2 1 1 4 2 4 %pi %pi %pi %pi sin(a + ---) - sqrt(2) sin(a - ---) sin(a - ---) sin(a - ---) 3 4 1 4 2 4 3 4 + 2 sin(a ) sin(a ) sin(a ) 1 2 3 = - sin(a ) sin(a ) sin(a ) + cos(a ) cos(a ) sin(a ) 1 2 3 1 2 3 + cos(a ) sin(a ) cos(a ) + sin(a ) cos(a ) cos(a ) 1 2 3 1 2 3
We can trade # of terms vs # of species by applying the new(?) formula recursively, e.g., a_1 = b_11+b_12+b_13, a_2= b_21+b_22+... . --rwg