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Piecewise Functions

Boole[expr]give 1 if expr is True, and 0 if it is False

Turning conditions into numbers.

Boole[expr] is a basic function that turns True and False into 1 and 0. It is sometimes known as the characteristic function or indicator function.
This gives the area of a unit disk.
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give the first vali for which condi is True
Piecewise[{{val1,cond1},...},val]give val if all condi are False

Piecewise functions.

It is often convenient to have functions with different forms in different regions. You can do this using Piecewise.
This plots a piecewise function.
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Piecewise functions appear in systems where there is discrete switching between different domains. They are also at the core of many computational methods, including splines and finite elements. Special cases include such functions as Abs, UnitStep, Clip, Sign, Floor and Max. Mathematica handles piecewise functions in both symbolic and numerical situations.
This generates a square wave.
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Here is the integral of the square wave.
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