Question

Find the Fourier transform, \(\hat{f}(k)\), of \(f(x)=e^{-2 x^{2}+x}\).

Short answer

The Fourier transform, \( \hat{f}(k) \), of the function \(f(x)=e^{-2 x^{2}+x}\) is given by \(\hat{f}(k) = e^{\pi^2 k^2 - \frac{1}{16}}\sqrt{\pi}\)

Step-by-step solution

Identify the function

Identify the function which is to be transformed which in this case is \(f(x)=e^{-2 x^{2}+x}\).

Recall the formula for Fourier Transform

The formula for the Fourier Transform is \(\hat{f}(k) = \int_{-\infty}^{\infty} f(x) e^{-2 \pi i k x} dx\). Here, \( \hat{f}(k) \) represents the Fourier transform of \( f(x) \), and \( k \) is the frequency variable.

Substitute the function into the Fourier Transform formula

Now, substitute \( f(x) = e^{-2 x^{2}+x} \) in the Fourier transform formula. It gives \(\hat{f}(k) = \int_{-\infty}^{\infty} e^{-2 x^{2}+x} e^{-2 \pi i k x} dx\). This creates a manageable integral.

Consolidate the exponentials

Combine the terms \(-2x^{2} + x\) and \(-2\pi ikx\) under one exponential to get an equation in the form of a quadratic identity which can be later simplified for the integral. It gives \(\hat{f}(k) = \int_{-\infty}^{\infty} e^{-2 x^{2}+(1-2\pi ik)x} dx\).

Identify the quadratic form

To integrate this exponential, it is ideal to identify a perfect square. This can be factored into \((x-\pi ik)^{2}\). After solving, the integrand matches the formula of integral of a Gaussian, \(\int_{-\infty}^{\infty} e^{-x^{2}} dx = \sqrt{\pi}\) and the Fourier Transform is calculated.

Final evaluation of Fourier Transform

After evaluating and simplifying, we get the Fourier Transform. \(\hat{f}(k) = e^{\pi^2 k^2 - \frac{1}{16}}\sqrt{\pi}\)

Key concepts

Gaussian Integral

The Gaussian integral is a fundamental concept in mathematics, especially in probability and statistics, representing the integral of the Gaussian function, commonly known as the bell curve. The Gaussian function is given by:

• \[ f(x) = e^{-x^2} \]

This integral over the entire real line results in a known and often useful result:

• \[ \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi} \]

This result is crucial because it also provides solutions for more complex integrals, like those encountered in many areas of physics.
During integral calculations involving exponential functions (with quadratic forms), the Gaussian integral simplifies these complex forms, demonstrating its centrality in analysis. For example, the Fourier Transform of Gaussian-related functions often involves Gaussian integrals or reductions thereof.
In our exercise, recognizing the exponential as a Gaussian allows us to factor, rearrange, and ultimately simplify the integral computation, leading to a neat analytical solution. This is vital for finding the Fourier transform, as seen in the provided exercise, where the expression matches the perfect square form crucial to the Gaussian integral approach.

Exponential Functions

Exponential functions appear frequently in both pure and applied mathematics. They often have the form:

• \[ f(x) = e^{ax}\]

where \(a\) is a constant. Exponential functions grow at a rate proportional to their current value, which leads to their applications in growth processes.
The beauty of exponential functions lies in their simplicity when it comes to differentiation and integration. This makes them particularly useful when solving differential equations or performing Fourier transforms.
In our case, the function being transformed, \( f(x)=e^{-2x^2 + x} \), involves exponential terms that represent a mixture of constant linear growth \(x\) and quadratic decay \(-2x^2 \). This combination is particularly challenging.
By consolidating the exponentials with the Fourier transform kernel, the function's complexity is reduced, aiding in further simplification using mathematical techniques such as completing the square, which aligns it to a Gaussian-like form, ready for integration.

Quadratic Forms

Quadratic forms involve expressions where variables are squared, taking the form:

• \[ ax^2 + bxy + cy^2 \]

They appear frequently in various mathematical contexts such as linear algebra and calculus.
The importance of quadratic forms in our exercise comes from the ability to simplify expressions into perfect square forms which are easier to work with, especially while evaluating integrals. A classic technique used here is completing the square. This involves rewriting an expression \(ax^2 + bx + c\) as a perfect square, which simplifies integration:

• \[ a \left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right) \]

In the context of the Fourier transform and our function \( e^{-2x^2 + (1-2\pi ik)x} \), completing the square permits us to convert the quadratic form into a Gaussian integrable form.
This process ends up turning a possibly daunting calculation into something manageably simple, utilizing tools like the Gaussian integral to ultimately solve the Fourier transform presented in the exercise.