Find the exponential Fourier transform of the given f(x) and...

Find the exponential Fourier transform of the given $f(x)$ and write $f(x)$ as a Fourier integral [that is, find $g(\alpha)$ in equation (12.2) and substitute your result into the first integral in equation $(12.2)]$ $$f(x)=\left\{\begin{array}{cl} \cos x, & -\pi / 2 < x < \pi / 2 \\ 0, & |x| > \pi / 2 \end{array}\right.$$ Hint: In Problems 11 and $12,$ use complex exponentials.

Find the exponential Fourier transform of the given $f(x)$ and write $f(x)$ as a Fourier integral [that is, find $g(\alpha)$ in equation (12.2) and substitute your result into the first integral in equation $(12.2)]$
$$f(x)=\left\{\begin{array}{cl}
\cos x, & -\pi / 2 < x < \pi / 2 \\
0, & |x| > \pi / 2
\end{array}\right.$$
Hint: In Problems 11 and $12,$ use complex exponentials.

Key Concepts

Complex Exponential Representation

This key concept makes use of Euler's formula, which expresses sine and cosine functions in terms of complex exponentials. This representation simplifies many operations in Fourier analysis by transforming trigonometric integrations into exponential ones, easing the computation of Fourier transforms and the manipulation of frequency components.

Fourier Transform Pair

A Fourier transform pair consists of a function and its Fourier transform, which are interconvertible via the Fourier transform and its inverse. This duality is central to Fourier analysis, as it provides a systematic method for analyzing the frequency content of functions and reconstructing the original function from its frequency representation.

Exponential Fourier Transform

This concept involves representing a function in terms of complex exponentials where the Fourier transform is defined as an integral of the product of the function and a complex exponential kernel. It converts the function from the time or spatial domain to the frequency domain, allowing analysis of its frequency components. The transform and its inverse form a Fourier transform pair that facilitates the reconstruction of the original function.

Fourier Integral Representation

This is the process of expressing a function as an integral over a continuum of frequency components. Instead of using a discrete Fourier series for periodic functions, a Fourier integral is used for functions defined on an infinite interval. It represents the function as a weighted superposition of complex exponentials, with the weights given by the Fourier transform.

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