I came across a certain integral, and couldn't solve it (I now know you have to use the classic t=tan(1/2)x to do so) but I had a go using some really horrible and blazé mathematics.
I've attacthed my efforts. Now I'd like someone to explain why on earth it should possibly work!!
Re: This really shouldn't work, but it does!! Why?
Why it shouldnt? A definite integral in the real space becomes an integral on the line
x(t) = t, y(t) = 0 (the real axis) in the complex space.
Although the typical way to solve this kind of integrals is through the transformation z = exp(ix) dz = izdx which gives an integral on the circle |z|=1. So I = 2pi*i*Sum(Res(zn)) where zn are the polars of the function inside the circle and Res is the intergration remainder (ok im not sure for these names, my language is not english)
Last edited by bilm_ks; Jul 7th, 2006 at 07:30 AM.
"bla, bla,... exists number M so for each n > M bla, bla..." Exists? Where is it? (Kronecker said...)
Re: This really shouldn't work, but it does!! Why?
Ah thanks that's a useful insight.
I only said it shouldn't work because 1) It's really heavy handed and takes no consideration of special cases or validity, and 2) It has a mistake in the first/second line!
u MUST be less than 1 for the substitution to make any sense, but later I let u=7/5. Also cos(x) does not equal sqrt(1-sin²(x)) it equals plus or minus sqrt(1-sin²(x)). I'm sure you can find many other horrific mamhematics within!
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