Chapter 7: Q21P (page 385)

Find the fourier transform of $f (x) = e^{- x^{2} / (2 σ^{2})}$ . Hint: Complete the square in the xterms in the exponent and make the change of variable $y = x + σ^{2} i α$ .Use tables or computer to evaluate the definite integral.

Short Answer

Expert verified

The fourier transform of $f (x) = e^{- x^{2} / (2 σ^{2})}$ is $g (α) = \frac{σ}{\sqrt{2 π}} e^{- σ^{2} α^{2} / 2}$ .

Step by step solution

Given Information.

The given function is $f (x) = e^{- x^{2} / (2 σ^{2})}$ .

Definition of fourier transform

The Fourier transform is a mathematical technique for expressing a function as the summation of sines and cosines functions.

Step 3: To find the fourier transform of the given function

The fourier transform is given below.

$g (α) = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{- x^{2} / (2 σ^{2})} e^{- i α x} d x$

Put $u = \frac{x}{σ \sqrt{2}}$

Differentiate with respect to x.

$d u = \frac{d x}{σ \sqrt{2}}$

$\begin{array}{l} g (α) = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{- u^{2} - i α \sqrt{2} σ u} \sqrt{2} σ d u \\ g (α) = \frac{σ}{π \sqrt{2}} \int_{- \infty}^{\infty} e^{- u^{2} - i α \sqrt{2} σ u} d u \\ g (α) = \frac{σ}{π \sqrt{2}} \int_{- \infty}^{\infty} e^{- {(u + i α σ / (\sqrt{2}))}^{2} - σ^{2} α^{2} / 2} d u \end{array}$

Simplify further

$g (α) = \frac{σ}{π \sqrt{2}} e^{- σ^{2} α^{2} / 2} \int_{- \infty}^{\infty} e^{- {(u + i α σ / (\sqrt{2}))}^{2}} d (u + i σ α / \sqrt{2})$

Put $u + i σ α / \sqrt{2} = η$

$g (α) = \frac{σ}{π \sqrt{2}} e^{- σ^{2} α^{2} / 2} \int_{- \infty}^{\infty} e^{- η^{2}} d η$

But the integral $\int_{- \infty}^{\infty} e^{- η^{2}} d η$ is Euler-poisson integral and its value is $\sqrt{π}$ .

$g (α) = \frac{σ}{\sqrt{2 π}} e^{- σ^{2} α^{2} / 2}$

The fourier transform of $f (x) = e^{- x^{2} / (2 σ^{2})}$ is $g (α) = \frac{σ}{\sqrt{2 π}} e^{- σ^{2} α^{2} / 2}$ .

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Find the fourier transform of $f (x) = e^{- x^{2} / (2 σ^{2})}$ . Hint: Complete the square in the xterms in the exponent and make the change of variable $y = x + σ^{2} i α$ .Use tables or computer to evaluate the definite integral.

Short Answer

Step by step solution

Given Information.

Definition of fourier transform

Step 3: To find the fourier transform of the given function

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Find the fourier transform off(x)=e−x2/(2σ2). Hint: Complete the square in the xterms in the exponent and make the change of variabley=x+σ2iα .Use tables or computer to evaluate the definite integral.

Short Answer

Step by step solution

Given Information.

Definition of fourier transform

Step 3: To find the fourier transform of the given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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Find the fourier transform of $f (x) = e^{- x^{2} / (2 σ^{2})}$ . Hint: Complete the square in the xterms in the exponent and make the change of variable $y = x + σ^{2} i α$ .Use tables or computer to evaluate the definite integral.