What properties of this complex function can be deduced? Let $\Phi(x,y)$ be a complex function of $x, y \in \mathbb{R}$ .
Given that $$\int_{-\infty}^{+\infty} \Phi^*(x,y) \frac{\partial}{\partial x} \Phi(x,y) \,dx$$
is purely imaginary:


*

*What properties of the complex function $\Phi(x,y)$ can be deduced?

 A: Hint: Note that
$$
\frac{\partial}{\partial x} |\Phi|^2 = 
\frac{\partial}{\partial x} (\Phi\Phi^*) = 
\Phi^*\frac{\partial \Phi}{\partial x} +  \Phi\frac{\partial \Phi^*}{\partial x} = 2 \operatorname{Re}\left[\Phi^*\frac{\partial \Phi}{\partial x} \right]
$$
A: Am not sure if this line of reasoning is correct. Omnomnomnom says that the answer I arrived at is correct in the comments. But, I wanted to make sure my reasoning is right. This is how I develepod on the hint provided by Omnomnomnom's answer:
If the integral of a complex function over some limits is purely imaginary, then the integral of the real part of the complex function over the same limits must be zero. Now, from the hint:
$$
\frac{\partial}{\partial x} |\Phi|^2 = 
\frac{\partial}{\partial x} (\Phi\Phi^*) = 
\Phi^*\frac{\partial \Phi}{\partial x} +  \Phi\frac{\partial \Phi^*}{\partial x} = 2 \operatorname{Re}\left[\Phi^*\frac{\partial \Phi}{\partial x} \right]
$$
So, putting all of this together, we get:
$$
\int_{-\infty}^{+\infty}\frac{\partial}{\partial x} |\Phi|^2 dx = 0
$$
Or:
$$
\lim_{x\to +\infty}|Φ(x,y)|^2 - \lim_{x\to −\infty}|Φ(x,y)|^2 = 0
$$
Further:
$$
\lim_{x\to +\infty}|Φ(x,y)|^2 = \lim_{x\to −\infty}|Φ(x,y)|^2
$$
