Find the Fourier cosine transform of e^-2x^2

The Fourier cosine transform \( \mathcal{F}_c\{f(x)\} \) of a function \( f(x) \) is defined as:

\[
\mathcal{F}_c\{f(x)\} = \int_{0}^{\infty} f(x) \cos(kx) \, dx
\]

For the function \( f(x) = e^{-2x^2} \), the Fourier cosine transform is:

\[
\mathcal{F}_c\{e^{-2x^2}\} = \int_{0}^{\infty} e^{-2x^2} \cos(kx) \, dx
\]

This integral can be evaluated using a known result:

\[
\int_{0}^{\infty} e^{-ax^2} \cos(bx) \, dx = \frac{1}{2} \sqrt{\frac{\pi}{a}} e^{-\frac{b^2}{4a}}
\]

In our case, \( a = 2 \) and \( b = k \), so the Fourier cosine transform becomes:

\[
\mathcal{F}_c\{e^{-2x^2}\} = \frac{1}{2} \sqrt{\frac{\pi}{2}} e^{-\frac{k^2}{8}}
\]

Thus, the Fourier cosine transform of \( e^{-2x^2} \) is:

\[
\mathcal{F}_c\{e^{-2x^2}\} = \frac{\sqrt{\pi}}{2\sqrt{2}} e^{-\frac{k^2}{8}}
\]

Would you like more details on any part of this process?

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Here are some related questions:

1. What is the significance of the Fourier cosine transform in signal processing?
2. How does the Fourier sine transform differ from the cosine transform?
3. Can you find the inverse Fourier cosine transform of \( \frac{\sqrt{\pi}}{2\sqrt{2}} e^{-\frac{k^2}{8}} \)?
4. How would the result change if the exponential function were \( e^{-ax^2} \) with a different \( a \)?
5. What is the physical interpretation of the Fourier transform in this context?
6. How does the Fourier transform relate to the Laplace transform?
7. What are some applications of the Fourier cosine transform in engineering?
8. Can you derive the integral used to evaluate the transform directly?

**Tip:** When working with Fourier transforms, always consider the function's symmetry properties—they can simplify your calculations significantly.

Explore how to find the Fourier cosine transform of e^-2x^2 using integral techniques and Fourier transform properties. Understand the mathematical concepts of Fourier cosine transforms and their applications. Suitable for advanced mathematics enthusiasts and students.

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