Difficult integral involving exponential

In summary, the conversation is about verifying the Fourier transform using the given pair of functions. The user is stuck on the integration and is attempting to solve it using integration by parts. Another user suggests completing the square and finding the values of alpha, beta, and gamma. There is also a discrepancy in the given formula for the Fourier transform.
  • #1
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0

Homework Statement


I'm trying to verify the Fourier transform but am getting stuck on the integration. Here is the pair:
[tex]f(x) = e^{-ax^2}[/tex]
[tex]\hat{f}(k) = \frac{1}{\sqrt{2a}}e^{-k^2/4a}[/tex]
[tex]a>0[/tex]

Homework Equations



I know that
[tex]\hat{f}(k)=\int_{-\infty}^{\infty}f(x)e^{ikx}dx[/tex]

The Attempt at a Solution



So I have
[tex]\hat{f}(k)=\int_{-\infty}^{\infty}e^{-ax^2}e^{ikx}dx[/tex]
[tex]\hat{f}(k)=\int_{-\infty}^{\infty}e^{-ax^2+ikx}dx[/tex]

I tried using integration by parts and I'm not sure that's the right way to go. If it is I'm not sure how to go about it without getting a more complicated integral.
 
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  • #2
You need to complete the square, which means:

[tex]-(ax^2+ikx)=-\left[(\alpha x+\beta)^2+\gamma \right][/tex].

Find [itex]\alpha,\beta[/itex] and [itex]\gamma[/itex].

Edit1: it seems you have either listed [itex]\hat{f}(k)[/itex] wrong or the book where you got [itex]\hat{f}(k)[/itex] from is wrong, because the answer should be:

[tex]
\hat{f}(k) = \sqrt{\frac{\pi}{a}}e^{-k^2/4a}
[/tex]

edit2: While making no difference to the final answer in this case, shouldn't it be [tex]\hat{f}(k)=\int_{-\infty}^{\infty}f(x)e^{-ikx}dx[/tex], note the minus sign.
 
Last edited:

1. What is an integral involving exponential?

An integral involving exponential is a mathematical expression that involves the exponential function, which is a function of the form f(x) = ex, where e is a mathematical constant approximately equal to 2.71828. The integral involves finding the area under the curve of the exponential function between two specified limits.

2. Why are integrals involving exponential considered difficult?

Integrals involving exponential are considered difficult because they cannot be solved using basic integration techniques such as substitution or integration by parts. They require more advanced methods such as integration by partial fractions, trigonometric substitution, or series expansion.

3. How do you solve a difficult integral involving exponential?

There is no one universal method for solving difficult integrals involving exponential. The approach will depend on the specific form of the integral. Some common techniques include integration by parts, substitution, partial fractions, and series expansion. It may also be helpful to use a computer or calculator to approximate the integral.

4. What is the purpose of solving a difficult integral involving exponential?

The purpose of solving a difficult integral involving exponential is to determine the exact value of the integral, which can be useful in various mathematical and scientific applications. For example, it can be used to calculate probabilities in statistics, to solve differential equations in physics and engineering, and to find areas and volumes in geometry.

5. Are there any tips for solving difficult integrals involving exponential?

Yes, there are a few tips that can make solving difficult integrals involving exponential easier. These include breaking the integral into smaller parts, using symmetry to simplify the integral, and using properties of the exponential function such as e-x = 1/ex and ex * ey = ex+y. It can also be helpful to practice and familiarize yourself with different integration techniques.

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