Your equation is blatantly wrong - ln(a) (a E |R) isn''t defined for a <= 0.
The Euler angle formula is e^(i*phi) = cos(phi) + i * sin(phi).
cosine without pi
My favourite way of computing Pi is based on the curious fact that the probability of two random integers being relatively primes is P = 6/Pi^2 (I didn''t smoke anything weird). So you can just estimate that probability by generating a bunch of random integers and checking if they are relatively prime. The last step is Pi = sqrt(6/P).
quote:
Your equation is blatantly wrong - ln(a) (a E |R) isn''t defined for a <= 0.
Not exactly true. In complex analysis, log is a multi-valued function, and the possible values of ln(-1) are (2*k-1)*i*Pi, where k is an integer, so the formula is right if you select the appropriate branch of logarithm in that region.
Yep, I know.
Sorry, I thought you meant ln: |R -> |R. Gotta write -1 E C or (-1, 0), or I''ll go making false assumptions
Sorry, I thought you meant ln: |R -> |R. Gotta write -1 E C or (-1, 0), or I''ll go making false assumptions
quote:Original post by Jan Wassenberg
Sorry, I thought you meant ln: |R -> |R. Gotta write -1 E C or (-1, 0), or I''ll go making false assumptions
I thought the "i" make it quite clear what he meant...
pi is something like 4*(1/2 + 1/3 - 1/4 + 1/5 - 1/6 ...)
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Here's a few...
π = 22/7
π/2 = (2 × 2 × 4 × 4 × 6 × 6 × 8 × … ) / (1 × 3 × 3 × 5 × 5 × 7 × 7 × … )
π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - …
π/6 = 1/2 + 1 / 2(3 × 23) + (1 × 3) / (2 × 4)(5 × 25) + (1 × 3 × 5) / (2 × 4 × 6)(7 × 27) + …
π = 2√3 * (1 - (3 * 3)^-1 + (3^2 * 5)^-1 - (3^3 * 7)^-1 + … )
π^3 / 8 = ∫ (log x)2 / (1 + x2) dx from 0 to ∞
[edited by - Dracoliche on October 10, 2002 12:05:59 PM]
π = 22/7
π/2 = (2 × 2 × 4 × 4 × 6 × 6 × 8 × … ) / (1 × 3 × 3 × 5 × 5 × 7 × 7 × … )
π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - …
π/6 = 1/2 + 1 / 2(3 × 23) + (1 × 3) / (2 × 4)(5 × 25) + (1 × 3 × 5) / (2 × 4 × 6)(7 × 27) + …
π = 2√3 * (1 - (3 * 3)^-1 + (3^2 * 5)^-1 - (3^3 * 7)^-1 + … )
π^3 / 8 = ∫ (log x)2 / (1 + x2) dx from 0 to ∞
[edited by - Dracoliche on October 10, 2002 12:05:59 PM]
drac: sorry, but according to windows calculator 22/7 is out by 0.00126448926734961868021375957764
quote:
I don''t care what pi specificaly is, and I know how to use the windows calculator. I just thought it would be cool to have an equation that could potentialy calculate pi exactly, even though it would take an infinite amount of time.
The sum from n = 0 to INF of 1/n^2 is Pi^/6. Calculate the nth partial sum, multiply by 6, and take the square root. There you have an approximation to Pi. If you find a book on Fourier analysis you can find a bunch of other infinite series that converge to Pi.
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