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.

One App. One Place for Learning.

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