Chapter 7: Q8P (page 384)

Find the exponential Fourier transform of the given $f (x)$ and write $f (x)$ as a Fourier integral.
$f (x) = {\begin{cases} x, 0 < x < 1 \\ 0, o t h e r w i s e \end{cases}$

Short Answer

Expert verified

Answer

The exponential Fourier transform of the given function is $g (α) = \frac{1 e^{+ i α} (1 + i α) - 1}{α^{2}}$ and $f (x)$ as a Fourier integral is $f (x) = \frac{1}{2 π} \int_{- \infty}^{\infty} \frac{e^{- i α} (1 + i α) - 1}{α^{2}} e^{i α x} d α$ .

Step by step solution

Given information

The given function is $f (x) = \{\begin{cases} x, 0 < x < 1 \\ 0, o t h e r w i s e \end{cases}$ . The exponential Fourier transform of the function is to be found and the function is to be written as a Fourier integral.

The significance of Fourier transforms

The following are the formulas for the Fourier series transforms.

Here,is called the Fourier transform of.

Find the Fourier transform

Use the formula $g (α) = \frac{1}{2 π} \int_{- \infty}^{\infty} f (x) e^{- i α x} d x$ to find the Fourier tr $g (α) = \frac{1}{2 π} \int_{- \infty}^{\infty} f (x) e^{- i α x} d x = \frac{1}{2 π} {[\frac{x}{i α} e^{- i α x} + \frac{1}{α^{2}} e^{- i α x}]}_{0}^{1} = \frac{1}{2 π} [\frac{e^{- i α}}{i α} + \frac{1}{α^{2}} (e^{- i α} - 1)] = \frac{1}{2 π} \frac{e^{- i α} - 1 + i α e^{- i α}}{α^{2}}$

Solve further to obtain.

Now, writeas a Fourier integral using the formula.

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Find the exponential Fourier transform of the given $f (x)$ and write $f (x)$ as a Fourier integral.
$f (x) = {\begin{cases} x, 0 < x < 1 \\ 0, o t h e r w i s e \end{cases}$

Short Answer

Step by step solution

Given information

The significance of Fourier transforms

Find the Fourier transform

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Find the exponential Fourier transform of the given f(x)and write f(x)as a Fourier integral.f(x)={x,0<x<10,otherwise

Short Answer

Step by step solution

Given information

The significance of Fourier transforms

Find the Fourier transform

One App. One Place for Learning.

Most popular questions from this chapter

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Find the exponential Fourier transform of the given $f (x)$ and write $f (x)$ as a Fourier integral.
$f (x) = {\begin{cases} x, 0 < x < 1 \\ 0, o t h e r w i s e \end{cases}$