Derive Formula 7 Using Formula 1 with Fourier Transforms

Math Problem Statement

فرمول 7 را با استفاده از فرمول 1 بدست بیار

To proceed with deriving formula 7 using formula 1, I'll first identify the relevant components from both formulas.

Formula 1 represents a step function:

1 & \text{if } -b < x < b \\ 0 & \text{otherwise} \end{cases}$$ **Formula 7** seems to involve the Fourier transform of a function involving an exponential, specifically: $$f(w) = \frac{i}{\sqrt{2\pi}} \left( e^{ibw} - e^{i(a-w)} \right)$$ To connect the two, we should apply the Fourier transform to a piecewise function. The key idea is to express the Fourier transform of a piecewise function like formula 1. Given the step-like nature of the function in Formula 1, it should be possible to represent this using the Fourier transform, and thereby derive a corresponding formula. Would you like me to elaborate on these steps further or clarify any specific parts of the transformation process?

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Fourier Transform
Piecewise Functions
Exponential Functions

f(x) = { 1 if -b < x < b, 0 otherwise }
F(w) = (i / sqrt(2π)) (e^(ibw) - e^(i(a-w)))

Fourier Transform Properties

Undergraduate Level

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