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Evaluate: $$\int_C \frac{e^z+\sin{z}}{z}dz$$ where, $C$ is the circle $|z|=5$ traversed once in the counterclockwise direction.

I can't find an anti-derivative to this function, and I am not sure one exists. I was thinking about using several theorems but each come up short. For instance, I cannot use the Closed Curve Theorem since the function in the integrand is not entire (at least I don't think it is).

Any hints (to help get me started)?

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    $\begingroup$ For example, you could use Cauchy's integral formula $\endgroup$ Feb 21, 2015 at 19:32
  • $\begingroup$ But the function is not entire, so how can I use Cauchy's Integral Formula? $\endgroup$ Feb 21, 2015 at 19:40
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    $\begingroup$ The numerator is entire, and that's all you need for the formula. Don't confuse this with Cauchy's integral theorem, which is specifically about the integral of entire functions. $\endgroup$ Feb 21, 2015 at 19:42
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    $\begingroup$ Ah! Thank you for that last comment. I get it now. $\endgroup$ Feb 21, 2015 at 19:45

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Let $f(z) = e^z + \sin z$, then by Cauchy's Integral Formula we have

$$2\pi i f(0) = \int_C \frac{e^z + \sin z }{z - 0} dz \implies \int_C \frac{e^z + \sin z }{z - 0} dz = \color{#f03}{ 2\pi i} $$

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