Find the Fourier sinw and cosine transform of x to the power n-1, n>0

To find the Fourier sine and cosine transforms of \( x^{n-1} \) where \( n > 0 \), we can use the definitions of the Fourier sine and cosine transforms.

### Fourier Sine Transform

The Fourier sine transform \( \mathcal{F}_s \{ f(x) \} \) of a function \( f(x) \) is defined as:
\[ \mathcal{F}_s \{ f(x) \}(\omega) = \sqrt{\frac{2}{\pi}} \int_0^\infty f(x) \sin(\omega x) \, dx \]

For \( f(x) = x^{n-1} \), we have:
\[ \mathcal{F}_s \{ x^{n-1} \}(\omega) = \sqrt{\frac{2}{\pi}} \int_0^\infty x^{n-1} \sin(\omega x) \, dx \]

Using the integral:
\[ \int_0^\infty x^{\mu} \sin(\beta x) \, dx = \frac{\Gamma(\mu + 1) \sin\left( \frac{(\mu + 1) \pi}{2} \right)}{\beta^{\mu + 1}}, \quad \text{for} \ \mu > -1 \]

Setting \( \mu = n-1 \) and \( \beta = \omega \), we get:
\[ \int_0^\infty x^{n-1} \sin(\omega x) \, dx = \frac{\Gamma(n) \sin\left( \frac{n \pi}{2} \right)}{\omega^n} \]

Thus, the Fourier sine transform is:
\[ \mathcal{F}_s \{ x^{n-1} \}(\omega) = \sqrt{\frac{2}{\pi}} \cdot \frac{\Gamma(n) \sin\left( \frac{n \pi}{2} \right)}{\omega^n} \]

### Fourier Cosine Transform

The Fourier cosine transform \( \mathcal{F}_c \{ f(x) \} \) of a function \( f(x) \) is defined as:
\[ \mathcal{F}_c \{ f(x) \}(\omega) = \sqrt{\frac{2}{\pi}} \int_0^\infty f(x) \cos(\omega x) \, dx \]

For \( f(x) = x^{n-1} \), we have:
\[ \mathcal{F}_c \{ x^{n-1} \}(\omega) = \sqrt{\frac{2}{\pi}} \int_0^\infty x^{n-1} \cos(\omega x) \, dx \]

Using the integral:
\[ \int_0^\infty x^{\mu} \cos(\beta x) \, dx = \frac{\Gamma(\mu + 1) \cos\left( \frac{(\mu + 1) \pi}{2} \right)}{\beta^{\mu + 1}}, \quad \text{for} \ \mu > -1 \]

Setting \( \mu = n-1 \) and \( \beta = \omega \), we get:
\[ \int_0^\infty x^{n-1} \cos(\omega x) \, dx = \frac{\Gamma(n) \cos\left( \frac{n \pi}{2} \right)}{\omega^n} \]

Thus, the Fourier cosine transform is:
\[ \mathcal{F}_c \{ x^{n-1} \}(\omega) = \sqrt{\frac{2}{\pi}} \cdot \frac{\Gamma(n) \cos\left( \frac{n \pi}{2} \right)}{\omega^n} \]

### Summary

- Fourier Sine Transform:
  \[ \mathcal{F}_s \{ x^{n-1} \}(\omega) = \sqrt{\frac{2}{\pi}} \cdot \frac{\Gamma(n) \sin\left( \frac{n \pi}{2} \right)}{\omega^n} \]

- Fourier Cosine Transform:
  \[ \mathcal{F}_c \{ x^{n-1} \}(\omega) = \sqrt{\frac{2}{\pi}} \cdot \frac{\Gamma(n) \cos\left( \frac{n \pi}{2} \right)}{\omega^n} \]

Would you like any further details or have any questions?

Here are some additional questions you might want to explore:

1. What is the Laplace transform of \( e^{-at} \sin(bt) \)?
2. How do you find the inverse Fourier transform of a given function?
3. What are the properties of the Fourier transform?
4. How is the Fourier transform used in signal processing?
5. Can you explain the relationship between the Fourier series and the Fourier transform?

**Tip:** When dealing with transforms, remember that understanding the properties and common pairs can significantly simplify the process of finding transforms for more complex functions.

Explore the Fourier sine and cosine transforms of the function x^(n-1) where n > 0. Understand how these transforms are computed using integral definitions and the gamma function. Suitable for undergraduate mathematics students.

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