i am currently going through a course about complex functions. when i was solving some home exercises about the Cauchy's integral theorem, i noticed something i just cant find the answer to.
It goes like this:
It is easy to see that the answer to the following integral is like so: Using Cauchy's integral theorem
$\int \frac{e^{z}}zdz = 2i\pi\ $ - through any rectifiable path
If i change parameters and say that
$ z=e^{i\theta} $ so that,
$dz=ie^{i\theta}d\theta$
And now i replace the parameters inside the integral so i get
$\int \frac{e^{z}}zdz = \int \frac{e^{e^{i\theta}}}{e^{i\theta}}ie^{i\theta}d\theta = 2i\pi\ $
i can now cancel the $ e^{i\theta} $
$\int {e^{e^{i\theta}}}id\theta = 2i\pi\ $
Now i can divide by $i$ on both sides and i get the final result:
$\int {e^{e^{i\theta}}}d\theta = 2\pi\ $
So, my question is how suddenly the result of the integral became a Real number. when first it was clearly a Complex number. Was there something wrong in the process of the parameter replacement or there is some deeper explanation to this.
Thank you.
What did you expect? You have divided by $i$! Yes, the original integral is $\int_\gamma\frac{e^{iz}}z\,\mathrm dz$, which, yes, is equal to $2\pi i$. But after dividing by $i$ what you get is$$\frac1i\int_\gamma\frac{e^{iz}}z\,\mathrm dz=2\pi,$$and $2\pi$ is a real number.