Fourier Sin Transform of $x^{m-1}$: Definition

This question asks us to identify the correct integral representation for the Fourier sin transform applied to the function $f(x) = x^{m-1}$. We need to carefully examine the components of each option provided, focusing on the function, the sine term, and the integration limits.

Integral Transforms Overview

Integral transforms are powerful mathematical tools used to convert functions from one domain (like time or space) into another domain (like frequency). The Fourier transform and its variants, including the Fourier sin transform, are fundamental in areas such as signal processing, physics, and engineering.

Sin Transform Components and $x^{m-1}$

The Fourier sin transform is specifically concerned with the sine component of a signal's frequency spectrum. For a function $f(x)$, its Fourier sin transform typically involves an integral of $f(x)$ multiplied by $\sin(sx)$, where $s$ represents the frequency variable.

In this specific question, the function being transformed is $f(x) = x^{m-1}$. Therefore, the integrand will contain the term ${x^{m - 1}}\sin sx$. We must determine the correct limits for the integral based on the definition and the options provided.

Integral Limits and Transform Types

The integration limits are crucial for distinguishing between different types of transforms or definitions. Standard definitions of the Fourier sin transform are typically applied to functions defined over the positive real axis, i.e., $x \in [0, \infty)$. Options 1 and 3 reflect this convention:

• Option 1: \(\mathop \smallint \limits_0^\infty {x^{m - 1}}\sin sx\)
• Option 3: \(\mathop \smallint \limits_0^\infty {x^{m - 1}}\sin x\)

Options 2 and 4 use integration limits from $-\infty$ to $\infty$. While the standard Fourier sin transform usually integrates from 0 to $\infty$, the context of the provided options includes definitions over the entire real line.

Selecting the Correct Integral Form

We are looking for the Fourier sin transform of the function $x^{m-1}$. Let's analyze the options against the core components:

• The function is $x^{m-1}$.
• The sine component is $\sin(sx)$.
• The integration limits need to be determined from the options.

Option 2 is \(\mathop \smallint \limits_{ - \infty }^\infty {x^{m - 1}}\sin sx\). This option correctly incorporates the function $x^{m-1}$ and the frequency term $\sin(sx)$. Crucially, it uses the integration limits $(-\infty, \infty)$. Given the choices presented in the question, this specific integral form represents the Fourier sin transform of $x^{m-1}$ as required.