Limit of an Integral of a $C^\infty$-Smooth Function with Compact Support
Let $\varphi:\R\rightarrow\R$ be a $C^\infty$-smooth function with compact support. Prove that the following limit exists, and compute the limit.
$$\lim_{\varepsilon\rightarrow0+} \int_{-\infty}^{\infty}\frac{\varphi(x)}{x+i\varepsilon} \;\mathrm{d}x , i=\sqrt{-1}$$
Answer
- The questioner was satisfied and accepted the answer, or
- The answer was disputed, but the judge evaluated it as 100% correct.
1 Attachment
-
Solving and writing up the solution to this question took a few hours. Please consider setting the price at a more appropriate level depending on the question's difficulty.
-
Hello. Sorry about that. First time using the service, so I didn't know what a good price would be. I also didn't know it would be so difficult to solve - I thought it would mostly use basic definitions since the other questions on this exam use mostly definitions or key but simply-stated theorems like the extreme value theorem. I would be willing to up the price but I don't think there is a way to do it after the fact?
- answered
- 201 views
- $10.00
Related Questions
- A lower bound on infinite sum of exponential functions (corrected version)
- Use first set of data to derive a second set
- Prove that $A - B=A\cap B^c$
- Assume there is no $x ∈ R$ such that $f(x) = f'(x) = 0$. Show that $$S =\{x: 0≤x≤1,f(x)=0\}$$ is finite.
- real analysis
- Need Upper Bound of an Integral
- Prove that $\frac{d \lambda}{d \mu} = \frac{d \lambda}{d \nu} \frac{d \nu}{d \mu}$ for $\sigma$-finite measures $\mu,\nu, \lambda$.
- Generalization of the Banach fixed point theorem