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riddles >> putnam exam (pure math) >> Complex Sum
(Message started by: Sameer on Sep 11th, 2007, 11:27pm)

Title: Complex Sum
Post by Sameer on Sep 11th, 2007, 11:27pm
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 + ...

Title: Re: Complex Sum
Post by iyerkri on Sep 11th, 2007, 11:53pm
After a lot of questionable mathematics, I arrive at : [hideb]
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.....
[/hideb]

Title: Re: Complex Sum
Post by Eigenray on Sep 12th, 2007, 2:31am
[hide]Different branches of log[/hide] are not a problem here.  [hide](eix)k = eikx when k is an integer (but not in general!).  And -http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gif (-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)[/hide].

Title: Re: Complex Sum
Post by Barukh on Sep 15th, 2007, 8:26am
Extremely nice approach, iyerkri!  

:D

Title: Re: Complex Sum
Post by iyerkri on Sep 16th, 2007, 1:52pm
thanks. By the way, that was probably my first correct post on the forum!



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