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Topic: Complex Sum (Read 785 times) |
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Sameer
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Pie = pi * e
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Complex Sum
« on: Sep 11th, 2007, 11:27pm » |
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Going by Complex numbers and summation themes!! Trying to read up for solving these, I found an interesting problem from my Engineering Math book!!
Find the Sum of the series:
sin2x - (1/2)sin(2x)*sin2x + (1/3)sin(3x)*sin3x - (1/4)sin(4x)*sin4x + ...
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| « Last Edit: Sep 11th, 2007, 11:28pm by Sameer » |
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"Obvious" is the most dangerous word in mathematics.
--Bell, Eric Temple
Proof is an idol before which the mathematician tortures himself.
Sir Arthur Eddington, quoted in Bridges to Infinity
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iyerkri
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Re: Complex Sum
« Reply #1 on: Sep 11th, 2007, 11:53pm » |
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After a lot of questionable mathematics, I arrive at : | hidden: |
arctan ( (tan x)^2/(1 + tanx + (tan x)^2)).
I am not to able to simplify further. I considered a similar series where sin kx is replaced by coskx, added the two, getting a power series in exp(ix)sinx, which evaluates to log(1 + exp(ix)sinx) , whose imaginary part is of interest to us, which is the above expression.
obviously I ignore the many branches of log and all.....
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| « Last Edit: Sep 11th, 2007, 11:54pm by iyerkri » |
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Eigenray
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Re: Complex Sum
« Reply #2 on: Sep 12th, 2007, 2:31am » |
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Different branches of log are not a problem here. (eix)k = eikx when k is an integer (but not in general!). And - (-x)k/k = log(1+x), the principal branch, as long as |x|<1, which it is in this case (except when |sin(x)|=1, but this isn't a problem).
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| « Last Edit: Sep 12th, 2007, 2:54am by Eigenray » |
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Barukh
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Re: Complex Sum
« Reply #3 on: Sep 15th, 2007, 8:26am » |
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Extremely nice approach, iyerkri!
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iyerkri
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Re: Complex Sum
« Reply #4 on: Sep 16th, 2007, 1:52pm » |
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thanks. By the way, that was probably my first correct post on the forum!
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