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I'm getting ahead in my differential equations textbook (Fundamentals of Differential Equations by Nagle et. al) and in the chapter of Laplace Transforms it states that the rectangular window function $\Pi_{a,b}\left(t\right)$ is given by \begin{align} \Pi_{a,b}\left(t\right):=u\left(t-a\right)-u\left(t-b\right)=\begin{cases} 0, & t<a, \\ 1, & a<t<b, \\ 0, & b <t. \end{cases} \end{align} However, in another textbook I'm reading about Fourier Transforms (which I ATM know very little about, just the basics since I just got it, Fourier Transforms: an introduction for Engineers by Gray et. al.) they've stated that the rectangle function, a variation of the box function, considered by Bracewell (in The Fourier Transform and its applications, 1965) is given by \begin{align}\Pi\left(t\right)= \begin{cases} 1, & \left|t\right|<\frac{1}{2}, \\ \frac{1}{2}, & \left|t\right|=\frac{1}{2}, \\ 0, & \text{otherwise}. \end{cases} \end{align} Why are they so different yet have such similar names?

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With Fourier transforms and sampled band-limited signals the concept of a Dirac delta function and the Dirac comb sometimes comes up. They are zero everywhere else but at certain points. The integral over each of such special point equals one. Sometimes you need to multiply a Dirac comb by a rectangular function. If either of the edges coincides with a Dirac delta, then it matters what the value of the rectangular function is exactly at the edge. The multiplication results in scaling of the Dirac delta (and its integral) by that value. Setting the edge values to $\frac{1}{2}$ may give the most meaningful result, see: Whittaker-Shannon ($\operatorname{sinc}$) interpolation for a finite number of samples.

Where the rectangle ends is a matter of convention. It is better to define the rectangle function for the reader if you ever happen to use it.

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Other than the value of $1/2$ on the edges, they're not different, just different notation. I don't know the motivation for the $1/2$'s, since it's still discontinuous, but they presumably explain why in the text. If we use the second notation, then the first definition (ignoring the $1/2$'s) is just $$ \Pi \left( { t - (a+b)/2) } \over { a-b } \right). $$

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