I need to know what the meaning is, mathematically, of the phrase, f approaches zero when x approaches infinity, uniformly in a certain set.
For example, see Jordan Lemma.
What is mean for a function to be uniformly convergent?
Does it mean:
We say that f approaches 0, uniformly on the set E as x approaches infinity, if given an arbitrarily small positive number epsilon, there exist a number N that depends only on epsilon, such that when x is greater than N, f(x) is less than epsilon in absolute value for any x in E.
Is the above correct?
Here's an example from Basic Complex Analysis by Jerrold Marsden (1973). Page 235.
If $f(z)$ approaches $0$ when the absolute value of $z$ approaches $\infty$, uniformly in $\arg z$, and $\arg z$ between $0$ and $\pi$, then the contour integral over $\gamma(\rho)$ $$\oint_{\gamma(\rho)} \exp( i a z) f(z) \mathrm d z$$ approaches $0$ when $\rho$ approaches $\infty$. Here $\gamma(\rho)(\theta) = \rho \exp (i \theta)$ and $\theta$ between $0$ and $\pi$.