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If $C$ is the circle $|z|=3$

$$g(z)=\int_C \frac{2s^2-s-2}{s-z} ds$$

then using Cauchy Integral

$$g(2) =\int_C \frac{2s^2-s-2}{s-2} dz = 2\pi i (2(2^2)-2-2) = 8\pi i$$

But what can we say about the value of $g(z)$ when $|z| > 3$?

By Cauchy Goursat,

$$\int_C f(z) dz = 0$$

if $f(z)$ is analytic at all points interior to and on a simple closed contour C

but if

$$f(s) = \frac{2s^2-s-2}{s-z} $$

then we know that $f(s)$ is not analytic when $s=z$

so we know that there is no simple closed contour that can enclosed $s \neq z$

what am I missing here?

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  • $\begingroup$ For $\lvert z\rvert > 3$, the integrand $\frac{s^2-s-2}{s-z}$ has no singularities enclosed by $C$, so the integral is $0$. $\endgroup$ – Daniel Fischer Dec 14 '13 at 21:13
  • $\begingroup$ what if $z = 4$, then $f(s)$ is not analytic at $s=4$ $\endgroup$ – user2654176 Dec 14 '13 at 21:15
  • $\begingroup$ But that's outside the disk bounded by $C$. $\endgroup$ – Daniel Fischer Dec 14 '13 at 21:16
  • $\begingroup$ I apologize if this is trivial, but I thought $C$ is bound by $|z| > 3$, which is a lower bound for $z$ $\endgroup$ – user2654176 Dec 14 '13 at 21:19
  • $\begingroup$ $C$ is the boundary curve of the disk with radius $3$ centered at the origin. $z = 4$ lies outside the closure that disk. Thus $\frac{s^2-s-2}{s-4}$ is holomorphic in a neighbourhood of the disk. Cauchy's integral theorem says the integral is $0$. $\endgroup$ – Daniel Fischer Dec 14 '13 at 21:21
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Most of the confusion was probably caused by the switch from $z$ being a variable of integration in $\int f(z)\,dz$ to it being a fixed number in $$\int_C \frac{2s^2-s-2}{s-z} \,ds$$ The explanation given by Daniel Fischer is quoted to create an actual answer here:

  1. For $\lvert z\rvert > 3$, the integrand $\frac{s^2-s-2}{s-z}$ has no singularities enclosed by $C$, so the integral is $0$.

  2. For example, take $z=4$. Since $C$ is the boundary curve of the disk with radius $3$ centered at the origin, $z = 4$ lies outside the closure that disk. Thus $\frac{s^2-s-2}{s-4}$ is holomorphic in a neighbourhood of the disk. Cauchy's integral theorem says the integral is $0$.

  3. The contour is a closed path in the plane. $z$ is a parameter, nothing more. The function $f(s) = \frac{2s^2-s-2}{s-z}$ is (remember, $z$ is just a parameter) analytic in the entire plane minus the point $z$. If $\lvert z\rvert = 3$, then $f$ has a singularity on $C$, and the integral does not exist. If $\lvert z\rvert > 3$, the contour $C$ does not wind around the singularity of $f$, so by Cauchy's integral theorem the integral is $0$. If $\lvert z\rvert < 3$, the contour winds (once) around the singuarity of $f$, and the integral is (in general) nonzero, it is $2\pi i(2z^2-z-2)$ then.

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