Order of integration and taking limits

[dyn]

Hi.
I came across the following integral in contour integration
lim(ε→0) "integral of" exp(iaεe^iθ) dθ = θ
If I take the limit first then it just becomes the integral of 1 which is θ.

I have 2 questions -
If I take the limit first and then perform the integral do I always get the same answer as when I do the integral first and then take the limit ? In other words does the order of operations matter ?

If I wanted to do the integral first , how do I integrate the exponential of an exponential ? I have never seen one before.

Thanks

 

[fresh_42]

Hi.
I came across the following integral in contour integration
lim(ε→0) "integral of" exp(iaεe^iθ) dθ = θ
If I take the limit first then it just becomes the integral of 1 which is θ.

I have 2 questions -
If I take the limit first and then perform the integral do I always get the same answer as when I do the integral first and then take the limit ?

No.

In other words does the order of operations matter ?

Yes.
There are two helpful results for the real case in this context:
https://en.wikipedia.org/wiki/Dominated_convergence_theorem
https://en.wikipedia.org/wiki/Fatou's_lemma

If I wanted to do the integral first , how do I integrate the exponential of an exponential ? I have never seen one before.

It would probably be easier to prove that the conditions for an exchange of limit and integration are given.

 

[dyn]

Thanks for your reply. I am learning complex analysis from a physics background and to be honest most of those references were over my head

 

[fresh_42]

The best explanation I had found is

"Even if all ##f_n## and the limit function ##f## are integrable, the convergence of the integral sequence does in general not hold! The integrable major function (first link) prevents the sequence of functions ##f_n## from escaping to infinity."

This means we need something that dominates the limit sequence from above. As all your function values are on the unit circle, I guess this could be achieved in this case. And as a side note: physicists don't bother as long as the result matches their experiments, but do not quote me on that.