Chapter 7: Q7-13-17P (page 388)

Show that the Fourier sine transform of $x^{\frac{- 1}{2}}$ is . Hint: Make the change of variable $z = α x$ . The integral $\int_{0}^{\infty} z^{- 1 / 2} s i n z d z$ can be found by computer or in tables

Short Answer

Expert verified

Thus, the required Fourier series is $g (α) = \sqrt{\frac{2}{π}} α \frac{1}{2} \frac{π}{2}$

Step by step solution

Determine the Fourier transformation for the given function.

Fourier sine transform of f(x) is,

$g (α) = \sqrt{\frac{2}{π}} \int_{0}^{\infty} f (x) \sin α x d x = \sqrt{\frac{2}{π}} \int_{0}^{\infty} x^{\frac{1}{2}} \sin α x d x$

Let $z = α x, d z = α d x$

So, we get,

$g (α) = \sqrt{\frac{2}{π}} \int_{0}^{\infty} x^{\frac{1}{2}} \sin α x d x = \sqrt{\frac{2}{π}} \int_{0}^{\infty} {(\frac{z}{α})}^{\frac{1}{2}} \sin z \frac{d z}{α} = \sqrt{\frac{2}{π}} α^{\frac{1}{2}} \int_{0}^{\infty} \frac{\sin z}{z} d z g (α) = \sqrt{\frac{2}{π}} α^{\frac{1}{2}} \frac{π}{2}$

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Show that the Fourier sine transform of $x^{\frac{- 1}{2}}$ is . Hint: Make the change of variable $z = α x$ . The integral $\int_{0}^{\infty} z^{- 1 / 2} s i n z d z$ can be found by computer or in tables

Short Answer

Step by step solution

Determine the Fourier transformation for the given function.

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Show that the Fourier sine transform of x-12 is . Hint: Make the change of variablez=αx . The integral ∫0∞z-1/2sinzdzcan be found by computer or in tables

Short Answer

Step by step solution

Determine the Fourier transformation for the given function.

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Most popular questions from this chapter

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Show that the Fourier sine transform of $x^{\frac{- 1}{2}}$ is . Hint: Make the change of variable $z = α x$ . The integral $\int_{0}^{\infty} z^{- 1 / 2} s i n z d z$ can be found by computer or in tables