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I wanted to calculate:

$$\int\limits_{|z|=4} \frac{\sin(e^z)}{z} dz$$

Using the Cauchy integral form, it's easy to check that:

$$=\int\limits_{|z|=4} \frac{\sin e^z}{z-(0)} = 2\pi i \sin(e^0) = 2\pi i \sin(1) $$

But... then I thought, what if I compute the contour integral directly, without the use of Cauchy's theorem?
Evaluating the parametrization $z=4e^{i\theta}$:

$$ \begin{split} &=\int\limits_0^{2\pi} \frac{\sin\big(e^{4e^{i\theta}}\big)}{4e^{i\theta}} 4ie^{i\theta}d\theta\\ &=i\int\limits_0^{2\pi} \sin\big(e^{4e^{i\theta}}\big) d\theta \end{split} $$

and from there I'm stuck.

Any hint on the integral will be appreciated <3.

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  • $\begingroup$ What does $\dfrac{\sin(e^z)}z$ have to do with $\dfrac{e^z}{z+1}$? That said, I don't believe you (or anyone else) are going to evaluate that parametric integral directly. $\endgroup$ – Ted Shifrin Oct 13 '19 at 22:08
  • $\begingroup$ This is the big advantage of Cauchy integral formula, it helps us with integrals which are too hard to calculate in a direct way. So why would you try other ways? $\endgroup$ – Mark Oct 13 '19 at 22:12
  • $\begingroup$ Oops... that was a typo, thanks!. So, in summary, both of you think that there's no point on trying to evaluate the integral directly, right?. Thanks again for your time $\endgroup$ – holahola Oct 13 '19 at 22:41

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