Heaviside step function

The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument:

Related Topics:
Oliver Heaviside - Discontinuous - Function - Zero - One

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:u(x)=egin{cases} 0, & x < 0 \ 1, & x > 0 end{cases}

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The function is used in the mathematics of control theory and signal processing to represent a signal that switches on at a specified time and stays switched on indefinitely.

Related Topics:
Control theory - Signal processing

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It is the cumulative distribution function of a random variable which is almost surely 0. (See constant random variable.)

Related Topics:
Cumulative distribution function - Random variable - Almost surely - Constant random variable

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The Heaviside function is the integral of the Dirac delta function.

Related Topics:
Integral - Dirac delta function

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: u(x) = int_{-infty}^x { delta(t)} dt

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The value of u(0) is occasionally of disputed value. Some writers give u(0) = 0, some u(0) = 1. u(0) = 1/2 is the most consistent choice used, since it maximizes the symmetry of the function and becomes completely consistent with the signum function. This makes for a more general definition:

Related Topics:
Symmetry - Signum function

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: u(x) =

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egin{cases} 0, & x < 0

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\ rac{1}{2}, & x = 0

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\ 1, & x > 0

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end{cases}

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: u(x) = rac{1}{2} left ( 1 + sgn(x) ight )

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To remove the ambiguity of which value to use for u(0), a subscript specifying which value may be used:

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: u_n(x) =

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egin{cases} 0, & x < 0

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\ n, & x = 0

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\ 1, & x > 0

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end{cases}

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Often an integral representation of the step function is useful:

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:u(x)=lim_{ epsilon o 0} -{1over 2pi i}int_{-infty}^infty {1 over au+iepsilon} e^{-i x au} d au

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