Uniform convergence as $w$ goes to $\infty$

Question

If I have a function $f(z,w),z,w\in \mathbb{C}$ and I want to show $f(z,w)$ converges uniformly to $L(z)$ as $w \to \infty$, what should I prove?

I want to switch the limit and integral $\lim_{w\to \infty} \int f(z,w) dz =\int \lim_{w\to \infty} f(z,w) dz$. I know that we only need to verify uniform convergence. However, I am not sure about how to show uniform convergence in the case that $w$ is a complex number converging to infinity. I only know how to show uniform convergence in the case where $n$ is a natural number converging to infinity.

Answer 1

Basically the same:

$f(z,w) \rightrightarrows L(z)$ as $w \to \infty$ iff for each $\varepsilon >0$, there is some $K>0$, $|f(z,w) -L(z) |<\varepsilon$ whenever $z\in\mathbb C$ and $|w|>K$.

Let me know if i am wrong.