Evaluating $P\int_{-\infty}^{\infty}\exp(imx^2}dx$

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Homework Help Overview

The problem involves evaluating the integral $P\int_{-\infty}^{\infty}\exp(imx^2)dx$ for $m>0$. The context is within complex analysis, particularly focusing on contour integration techniques.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to define a contour for the integral and describes the segments of the contour. Some participants question the choice of contour and whether the integral approaches zero on certain segments. Others express uncertainty about applying the ML-bound to show that specific integrals vanish.

Discussion Status

The discussion is ongoing, with participants exploring the reasoning behind the contour choice and the implications for the integral's evaluation. There is a focus on understanding the behavior of the integral along different segments of the contour.

Contextual Notes

There is mention of a potential misunderstanding in the contour definitions, specifically regarding the coordinates used in segments C_2 and C_4. Participants are also navigating the lack of explicit equations or established methods for the problem.

Sam1234
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Homework Statement


Evaluate $P\int_{-\infty}^{\infty}\exp(imx^2}dx$, $m>0$.

Homework Equations


None


The Attempt at a Solution


Consider the following contour $C$ consisting of
$C_1:=\{z=x+iy:y=0, x:-s \to s\}$
$C_2:=\{z=x+iy:x=0, y:0 \to is\}$
$C_3:=\{z=x+iy:x: s \to -s, y: is \to -is \}$
$C_4:=\{z=x+iy:x=0: y:-is \to 0\}$

This is where i am stuck
 
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Sorry, i gave the wrong title
 
Well, why was that contour chosen? Does the integral go to 0 on some segments?
 
actually in C_2, x=s and in C_4, x=-s
 
Ok, I worked so far as that it only remains to show that the integrals on C_2 and C_4 are zero. I suspect having to use an ML-bound but don't really know how to do this.
 

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