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November 7th, 2012, 08:23 AM   #1
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Positively oriented circle...

{z:|z-z0|=p}.

Find ?_c_1(0) of (e^z + cosz)z^-1 dz

I know it should be 4pi(i)
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November 7th, 2012, 08:25 AM   #2
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Re: Positively oriented circle...

actually is Find ?_c_1(0) of (e(z) + cos(z))z^-1 dz
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December 4th, 2012, 08:14 AM   #3
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Re: Positively oriented circle...

The integral around a closed path (in this case a circle), is equal to times the sum of the residues at each pole of the integrand inside the circle. Here, everything is analytic except for that so the only pole is at z= 0 and that is a pole of order 1. I would expand as a Taylor's series around z= 0 and multiply each term by 1/z to get the Laurent series for the entire integrand. The residue is the coefficient of in the Laurent series.

Or you can do it directly. I presume that by "C_1(0)" is the unit circle with center at z= 0. A standard parameterization for the circle is so we can write the integral .
That looks difficult to me so I would use the "residue" method!
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December 6th, 2012, 12:14 AM   #4
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Re: Positively oriented circle...

[color=#000000]I guess you mean,



.
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