[math-fun] Weird arcsin pi formulas
ArcSin[(-1)^(1/4)/2^(3/4)] == Pi/8 + (1/2)*I*Log[1 + Sqrt[2]] (Compute just the real part?) Pi/2 == ArcSin[13/85] + ArcSin[5/13] + ArcSin[8/17] + ArcSin[33/65] (Obviously inferior to ArcSin[1/2].) Pi/2 == ArcSin[73/2665] + ArcSin[21/221] + ArcSin[13/85] + ArcSin[9/41] + ArcSin[7/25] + 2*ArcSin[5/13] Pi/2 == 4*ArcSin[191/18241] + 20*ArcSin[187/17485] - 4*ArcSin[157/12325] + 9*ArcSin[99/4901] + 8*ArcSin[93/4325] + 20*ArcSin[164/6725] + 8*ArcSin[160/6401] - ArcSin[140/4901] + 12*ArcSin[136/4625] --rwg asin(x)*asin(y) = sqrt(%pi)* inf ==== 2 2 2(n + 1) 2(n + 1) 2(n + 1) \ n! ((x sqrt(1-y ) + sqrt(1-x ) y) - y - x ) > ------------------------------------------------------------------ / 1 ==== 4 (n + 1) (n + -)! n = 0 2
[Can't believe I'm responding to a Gosper post . . .]
Pi/2 == ArcSin[13/85] + ArcSin[5/13] + ArcSin[8/17] + ArcSin[33/65]
This one is geometrically obvious, from four right triangles in the 2D lattice, following the sequence of points (0,56) (33,56) (48,36) (60,45) (84,13) (84,0) I wonder how many similar identities can be derived just from pulling right triangles from the lattice. -tom
Or as an arctangent identity: pi/2 = atan(+inf) = atan(13/84) + atan(5/12) + atan(8/15) + atan(33/56) or as a complex factorization pi/2 = Imag( log( (84+13I) (12+5I) (15+8I) (56+33I) )). The factorization looks hard-to-find, but can be built up from the complex factors of 5, 13, and 17 -- 2+-I, 3+-2I, 4+-I. Rich ------ Quoting Tom Rokicki <rokicki@gmail.com>:
[Can't believe I'm responding to a Gosper post . . .]
Pi/2 == ArcSin[13/85] + ArcSin[5/13] + ArcSin[8/17] + ArcSin[33/65]
This one is geometrically obvious, from four right triangles in the 2D lattice, following the sequence of points
(0,56) (33,56) (48,36) (60,45) (84,13) (84,0)
I wonder how many similar identities can be derived just from pulling right triangles from the lattice.
-tom
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Bill Gosper -
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Tom Rokicki