Integral of Complex Function

Calculus Level 3

Calculate $\displaystyle \oint_C\frac{5z^2-3z+2}{(z-1)^3}\, dz$ , where $C$ is a simple closed curve that contians the point $z=1$ .

10\pi

10\pi i

5\pi

5\pi i

1 solution

Aman Rajput
Jun 9, 2016

Use Cauchy Integral formula:

$\displaystyle \oint_C \frac{f(z)}{(z-a)^{n+1}} dz = f^n(a).\frac{2\pi i}{n!}$

Here , $f(z)$ is analytic since $|z|=1$ lies on the contour. Therefore , at $z=1$ and $n=2$ , we have

$\displaystyle \oint_C \frac{5z^2-3z+2}{(z-1)^{3}} dz = 10\pi i$

that's great

Andy Leonardo - 4 years, 12 months ago

The Cauchy Integral formula does not apply, since $z = 1$ lies on the contour $|z| = 1$ . The integral actually diverges.

Jon Haussmann - 4 years, 11 months ago

No it will not diverge.

Aman Rajput - 4 years, 11 months ago

@Aman Rajput Cauchy's integral formula only holds if the counter encloses the point, as opposed to the point lying on the counter. Previously, when it is stated "simple closed curve with $z = 1$ ", that is not clear what is being described.

Calvin Lin Staff - 4 years, 11 months ago

Integral of Complex Function

1 solution

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