I'm watching a video about the Jacobi Theta Function on YouTube.
At around 6:14 in the video, he has shown that...
$$\vartheta (x) = \sum_{n\in \mathbb Z} e^{-\pi {n^2} x} = \sum_{k\in \mathbb Z} {e^{-\pi {k^2} {1\over x}}} \int_{-\infty}^{\infty} {e^{-\pi x \big( y+ i {k\over x}\big)^2}} \space dy$$
I understand the steps he took to get to this point. But here is where I got confused.
He then looked at $y + i {k\over x}$ and he Let...
$y + i {k\over x} = z$
But then he said...
$dy = dz$
Because $i {k\over x}$ is not a complex variable.
I don't understand why it isn't. Isn't a complex number defined as $a+bi$ where $a,b\in \mathbb R$?
Am I missing something?
As $x$ is being held constant, the only thing varying in $z$ is $y$.