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Topic: Complex Sum (Read 778 times) 

Sameer
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Complex Sum
« on: Sep 11^{th}, 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: sin^{2}x  (1/2)sin(2x)*sin^{2}x + (1/3)sin(3x)*sin^{3}x  (1/4)sin(4x)*sin^{4}x + ...

« Last Edit: Sep 11^{th}, 2007, 11:28pm by Sameer » 
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iyerkri
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Re: Complex Sum
« Reply #1 on: Sep 11^{th}, 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..... 

« Last Edit: Sep 11^{th}, 2007, 11:54pm by iyerkri » 
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Eigenray
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Re: Complex Sum
« Reply #2 on: Sep 12^{th}, 2007, 2:31am » 
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Different branches of log are not a problem here. (e^{ix})^{k} = e^{ikx} 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).

« Last Edit: Sep 12^{th}, 2007, 2:54am by Eigenray » 
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Barukh
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Re: Complex Sum
« Reply #3 on: Sep 15^{th}, 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 16^{th}, 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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