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Contour Integrals
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Contour Integrals
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Topic: Contour Integrals (Read 7962 times)
ste
Newbie
Posts: 4
Contour Integrals
«
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?
Logged
Difficult takes a day. Impossible takes a week.
ste
Newbie
Posts: 4
Re: Contour Integrals
«
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
^{-1}
f(z
^{-1}
);0} = the residue of z
^{-1}
f(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
^{-1}
f(z
^{-1}
);0}) = 2(pi)i.
«
Last Edit: April 12, 2010, 01:38:09 PM by ste
»
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