Let the Fourier Integral Representation of F Is $f ( x ) = \int _ { 0 } ^ { \infty } [ A ( \alpha ) \cos ( \alpha x ) + B ( \alpha ) \sin ( \alpha x ) ] d \alpha / \pi$

question 3

Multiple Choice

Let $f ( x ) = \left\{ \begin{array} { c c c } 0 & \text { if } & x < - 1 \\1 & \text { if } & - 1 < x < 1 \\0 & \text { if } & x > 1\end{array} \right\}$ . The Fourier integral representation of f is $f ( x ) = \int _ { 0 } ^ { \infty } [ A ( \alpha ) \cos ( \alpha x ) + B ( \alpha ) \sin ( \alpha x ) ] d \alpha / \pi$ , where

• A) \(A ( \alpha ) = \sin \alpha / \alpha \text { and } B ( \alpha ) = \cos \alpha / \alpha\)
• B) \[A ( \alpha ) = 2 \cos \alpha / \alpha \text { and } B ( \alpha ) = 2 \sin \alpha / \alpha\]
• C) \(A ( \alpha ) = 2 \sin \alpha / \alpha \text { and } B ( \alpha ) = 2 \cos \alpha / \alpha\)
• D) \(A ( \alpha ) = 0 \text { and } B ( \alpha ) = 2 \sin \alpha / \alpha\)
• E) \(A ( \alpha ) = 2 \sin \alpha / \alpha \text { and } B ( \alpha ) = 0\)