# How to Solve a Contour Integral with exp(-z^2) Over a Rectangle?

• NT123
In summary, to integrate exp(-z^2) over the rectangle with vertices at 0, R, R + ia, and ia, we can use the integral of e^(-z^2) to evaluate the real integral int(0,inf)((e^(-x^2))*cos(2ax)), which is equal to sqrt(pi)*exp(-a^2)/2. This can be done by integrating e^(-z^2) over each of the four sides of the rectangle and writing out the entire contour integral. This will result in a non-analytic answer, but it will show the path of integration.
NT123

## Homework Statement

Integrate exp(-z^2) over the rectangle with vertices at 0, R, R + ia, and ia.

## Homework Equations

int(0, inf)(exp(-x^2)) = sqrt(pi/2)

## The Attempt at a Solution

I really don't have much of an idea here - the function is analytic so has no residues... The part from 0 to R is just the real integral, but for the other 3 sides I'm not too sure on how to proceed.

Isn't the contour integral equal to 0 if there are no poles?

ideasrule said:
Isn't the contour integral equal to 0 if there are no poles?
This is what I would have thought, but I'm supposed to be using the integral of e^(-z^2) to evaluate the real integral int(0,inf)((e^(-x^2))*cos(2ax)), which is apparently equal to
sqrt(pi)*exp(-a^2)/2.

Ah, that makes much more sense.

If we want to integrate from R+ia to ia, just integrate e^(-z^2)dz=e^-(x+ia)^2 dx from R to 0. Do the same for the other 3 sides. You won't get an analytic answer, but that's OK; just write out the entire contour integral first and you'll see where this is going.

## 1. What is a contour integral?

A contour integral is a type of line integral in complex analysis. It involves integrating a complex-valued function along a specific path or contour in the complex plane.

## 2. What is the purpose of a contour integral?

The purpose of a contour integral is to calculate the value of a complex integral by breaking it down into simpler integrals along a specific path in the complex plane. It is a useful tool in many areas of mathematics and physics, including complex analysis, number theory, and quantum mechanics.

## 3. How is a contour integral related to the Gaussian function exp(-z^2)?

A contour integral can be used to calculate the value of the Gaussian function exp(-z^2) along a specific path in the complex plane. This is known as the Gaussian integral and has important applications in probability, statistics, and physics.

## 4. What are some properties of the contour integral of exp(-z^2)?

The contour integral of exp(-z^2) is a special case of the error function and has a number of important properties. These include the fact that it is an even function, its value at z=0 is the square root of pi, and it can be used to calculate the probability of a standard normal distribution.

## 5. How is a contour integral of exp(-z^2) calculated?

The contour integral of exp(-z^2) can be calculated using various techniques, including the Cauchy integral theorem, Cauchy's integral formula, and the residue theorem. These methods involve evaluating the function along the chosen contour and using complex analysis techniques to simplify the integral.

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