Compute the integral of the Gaussian

In summary, Mr. Wolfram said that the inverse Fourier transform of the Fourier transform of ##f## is ##f##. We learned in class that the Fourier transform of a function is the inverse of its Fourier transform, so there should only be one Fourier transform. I'm uncertain of how to calculate this integral though. Mr. Wolfram showed me an indefinite integral involving an error function, but there has to be a different way to integrate it because we didn't learn about the error function.
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why does it say transforms? is there more than one Fourier transform?? we learned in class that the inverse Fourier transform of the Fourier transform of ##f## is ##f##, so there should be just one right? I'm uncertain of how to calulate this integral though.. Mr Wolfram showed me an indefinite integral involving an error function, but there has to be a different way to integrate it because we didn't learn about the error function. Mr Wolfram says

$$\int_{-\infty}^\infty fe^{-ikx}dx=\sqrt{\pi}\sigma e^{(\frac{\sigma k}{2})^2}e^{-ikx_0}$$

how does one go from the beginning to the finish?
 
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Write\begin{align*}
\tilde{f}(k) &= \int_{-\infty}^{\infty} \mathrm{exp}\left( -ikx - \frac{(x-x_0)^2}{\sigma^2} \right) dx \\
&= e^{-ikx_0} \int_{-\infty}^{\infty} \mathrm{exp}\left( -ik(x-x_0) - \frac{(x-x_0)^2}{\sigma^2} \right) dx\end{align*}from here a 'complete-the-square' substitution like ##u = \dfrac{x-x_0}{\sigma} + \dfrac{i\sigma k}{2}## looks helpful, but take some care to justify the limits of the integral over ##u## given that ##u## is complex.
 
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ergospherical said:
Write\begin{align*}
\tilde{f}(k) &= \int_{-\infty}^{\infty} \mathrm{exp}\left( -ikx - \frac{(x-x_0)^2}{\sigma^2} \right) dx \\
&= e^{-ikx_0} \int_{-\infty}^{\infty} \mathrm{exp}\left( -ik(x-x_0) - \frac{(x-x_0)^2}{\sigma^2} \right) dx\end{align*}from here a 'complete-the-square' substitution like ##u = \dfrac{x-x_0}{\sigma} + \dfrac{i\sigma k}{2}## looks helpful, but take some care to justify the limits of the integral over ##u## given that ##u## is complex.
\begin{align} \tilde{f}(k) =& \int_{-\infty}^{\infty} \mathrm{exp}\left( -ikx - \frac{(x-x_0)^2}{\sigma^2} \right) dx \\
=e^{-ikx_0} &\int_{-\infty}^{\infty} \mathrm{exp}\left( -ik(x-x_0) - \frac{(x-x_0)^2}{\sigma^2} \right)dx\\
=e^{-ikx_0} &\int_{-\infty}^{\infty} \mathrm{exp}\left( -\frac{\sigma^2i^2k^2}{4}+\frac{\sigma^2i^2k^2}{4}-ik(x-x_0) - \frac{(x-x_0)^2}{\sigma^2} \right)dx\\
=e^{-ikx_0-\frac{\sigma^2i^2k^2}{4}} &\int_{-\infty}^{\infty} \mathrm{exp}\left( -(\frac{\sigma i k}{2}+\frac{x-x_0}{\sigma})^2 \right)dx\end{align}
let ##u = \dfrac{x-x_0}{\sigma} + \dfrac{i\sigma k}{2}##. since ##u## is complex, it leads to integrating for ##u\in\{(-\infty,\infty)\times \{0\}\}##.
\begin{align}&\sigma e^{-ikx_0-\frac{\sigma^2i^2k^2}{4}}\int_ u e^{-u^2}du\\
=&\sigma e^{-ikx_0-\frac{\sigma^2i^2k^2}{4}} \Big[\frac{\sqrt{\pi}}{2}\text{erf}(u)\Big]_u\\
=&\sigma \sqrt{\pi} e^{-ikx_0+\frac{\sigma^2k^2}{4}}\end{align}
 
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  • #4
is it ok?
:)
 

What is the Gaussian function?

The Gaussian function, also known as the normal distribution, is a mathematical function that describes the probability distribution of a continuous random variable. It is commonly used in statistics, physics, and engineering to model real-world phenomena.

What does it mean to compute the integral of the Gaussian?

Computing the integral of the Gaussian means finding the area under the curve of the Gaussian function. This is done by evaluating the definite integral of the function over a given interval.

Why is computing the integral of the Gaussian important?

Computing the integral of the Gaussian is important because it allows us to calculate probabilities and make predictions about real-world data. It is also used in various fields of science, such as physics and chemistry, to solve complex equations and model natural phenomena.

How is the integral of the Gaussian calculated?

The integral of the Gaussian can be calculated using various methods, such as the trapezoidal rule or Simpson's rule. In some cases, it can also be solved analytically using integration techniques. There are also online calculators and software programs available for computing the integral of the Gaussian.

What are some applications of the Gaussian function?

The Gaussian function has many applications in science and engineering. It is used to model the distribution of values in a data set, such as the heights of a population or the scores on a test. It is also used in signal processing, image processing, and machine learning algorithms. Additionally, the Gaussian function is used in physics to describe the behavior of particles and in chemistry to model molecular motion.

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