wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

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

Title: Complex Sum
Post by Sameer on Sep 11^th, 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 11^th, 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 12^th, 2007, 2:31am

[hide]Different branches of log[/hide] are not a problem here. [hide](e^ix)^k = e^ikx 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 15^th, 2007, 8:26am

Extremely nice approach, iyerkri!

:D

Title: Re: Complex Sum
Post by iyerkri on Sep 16^th, 2007, 1:52pm

thanks. By the way, that was probably my first correct post on the forum!