Heaviside Step Function
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.
الجزء والصفحة:
...
26-9-2019
2519
Heaviside Step Function
The Heaviside step function is a mathematical function denoted 
, or sometimes 
or 
(Abramowitz and Stegun 1972, p. 1020), and also known as the "unit step function." The term "Heaviside step function" and its symbol can represent either a piecewise constant function or a generalized function.
When defined as a piecewise constant function, the Heaviside step function is given by
![H(x)=<span style=]() {0 x<0; 1/2 x=0; 1 x>0 " src="http://mathworld.wolfram.com/images/equations/HeavisideStepFunction/NumberedEquation1.gif" style="height:70px; width:115px" /> |
(1)
|
(Abramowitz and Stegun 1972, p. 1020; Bracewell 2000, p. 61). The plot above shows this function (left figure), and how it would appear if displayed on an oscilloscope (right figure).
When defined as a generalized function, it can be defined as a function 
such that
 |
(2)
|
for 
the derivative of a sufficiently smooth function 
that decays sufficiently quickly (Kanwal 1998).
The Wolfram Language represents the Heaviside generalized function as HeavisideTheta, while using UnitStep to represent the piecewise function Piecewise[![<span style=]()
{" src="http://mathworld.wolfram.com/images/equations/HeavisideStepFunction/Inline7.gif" style="height:15px; width:5px" />![<span style=]()
{" src="http://mathworld.wolfram.com/images/equations/HeavisideStepFunction/Inline8.gif" style="height:15px; width:5px" />1, x >= 0![<span style=]()
}" src="http://mathworld.wolfram.com/images/equations/HeavisideStepFunction/Inline9.gif" style="height:15px; width:5px" />![<span style=]()
}" src="http://mathworld.wolfram.com/images/equations/HeavisideStepFunction/Inline10.gif" style="height:15px; width:5px" />] (which, it should be noted, adopts the convention 
instead of the conventional definition 
).
The shorthand notation
 |
(3)
|
is sometimes also used.
The Heaviside step function is related to the boxcar function by
 |
(4)
|
and can be defined in terms of the sign function by
![H(x)=1/2[1+sgn(x)].](http://mathworld.wolfram.com/images/equations/HeavisideStepFunction/NumberedEquation5.gif) |
(5)
|
The derivative of the step function is given by
 |
(6)
|
where 
is the delta function (Bracewell 2000, p. 97).
The Heaviside step function is related to the ramp function 
by
 |
(7)
|
and to the derivative of 
by
 |
(8)
|
The two are also connected through
 |
(9)
|
where 
denotes convolution.
Bracewell (2000) gives many identities, some of which include the following. Letting 
denote the convolution,
 |
(10)
|
In addition,
The Heaviside step function can be defined by the following limits,
where 
is the erfc function, 
is the sine integral, 
is the sinc function, and 
is the one-argument triangle function. The first four of these are illustrated above for 
, 0.1, and 0.01.
Of course, any monotonic function with constant unequal horizontal asymptotes is a Heaviside step function under appropriate scaling and possible reflection. The Fourier transform of the Heaviside step function is given by
where 
is the delta function.
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.
Bracewell, R. "Heaviside's Unit Step Function, 
." The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 61-65, 2000.
Kanwal, R. P. Generalized Functions: Theory and Technique, 2nd ed. Boston, MA: Birkhäuser, 1998.
Spanier, J. and Oldham, K. B. "The Unit-Step 
and Related Functions." Ch. 8 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 63-69, 1987.
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