

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: sin^{2}x  (1/2)sin(2x)*sin^{2}x + (1/3)sin(3x)*sin^{3}x  (1/4)sin(4x)*sin^{4}x + ... 

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! 

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