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Contour Integrals
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Author Topic: Contour Integrals  (Read 7961 times)
ste
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« on: December 18, 2009, 06:42:27 PM »

Does anyone know how to integrate f(z-1)/z with respect to z over the clockwise oriented circle of radius 2, centred at the origin (of the complex plane), where f(w) = (w - cos(w))/exp(w) and z & w are complex numbers?
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« Reply #1 on: December 22, 2009, 02:20:30 PM »

I managed to work it out.

f(w) is entire because exp(w) is not equal to 0 for any w in the complex plane, however, (z-1)f(z-1) is not analytic at z = 0. Because the point z = 0 lies inside of the specified contour, we investigate the nature of this singularity by computing the Taylor series of f(z-1) about the point 0 and introducing a factor of z-1 to each term. Looking at the first term of the Taylor series, we see that res{z-1f(z-1);0} = the residue of z-1f(z-1) at 0 = -cos(0) = -1
 
By Cauchy's residue theorem the value of the integral is equal to -2(pi)i(res{z-1f(z-1);0}) = 2(pi)i.
« Last Edit: April 12, 2010, 01:38:09 PM by ste » Logged

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