What is Heaviside function: Definition and 51 Discussions

The Heaviside step function, or the unit step function, usually denoted by H or θ (but sometimes u, 1 or 𝟙), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive arguments. It is an example of the general class of step functions, all of which can be represented as linear combinations of translations of this one.
The function was originally developed in operational calculus for the solution of differential equations, where it represents a signal that switches on at a specified time and stays switched on indefinitely. Oliver Heaviside, who developed the operational calculus as a tool in the analysis of telegraphic communications, represented the function as 1.
The Heaviside function may be defined as:

a piecewise function:
an indicator function:
the derivative of the ramp function: The Dirac delta function is the derivative of the Heaviside function

Hence the Heaviside function can be considered to be the integral of the Dirac delta function. This is sometimes written as

although this expansion may not hold (or even make sense) for x = 0, depending on which formalism one uses to give meaning to integrals involving δ. In this context, the Heaviside function is the cumulative distribution function of a random variable which is almost surely 0. (See constant random variable.)
In operational calculus, useful answers seldom depend on which value is used for H(0), since H is mostly used as a distribution. However, the choice may have some important consequences in functional analysis and game theory, where more general forms of continuity are considered. Some common choices can be seen below.
Approximations to the Heaviside step function are of use in biochemistry and neuroscience, where logistic approximations of step functions (such as the Hill and the Michaelis–Menten equations) may be used to approximate binary cellular switches in response to chemical signals.

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  1. A

    Fourier sine and cosine transforms of Heaviside function

    Hi, I am really struggling with the following problem on the Fourier sine and cosine transforms of the Heaviside unit step function. The definitions I have been using are provided below. I tried each part of the problem, but I'm only left in terms of limits as x -> infinity of sin or cos...
  2. M

    Indefinite Integration of Heaviside function muliplied by a function

    Will it be [{(r-a)/r}*H(r-a)]
  3. F

    Charge invariance with Heaviside's function

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  4. B

    Why is the heaviside function in the inverse Laplace transform of 1?

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  5. CCMarie

    A Multi-variable function depending on the Heaviside function

    How can I calculate ∂/∂t(∫01 f(x,t,H(x-t)*a)dt), where a is a constant, H(x) is the Heaviside step function, and f is I know it must have something to do with distributions and the derivative of the Heaviside function which is ∂/∂t(H(t))=δ(x)... but I don't understand how can I work with the...
  6. C

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  7. B

    I Heavyside step function chain rule

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  8. Sudhir Regmi

    I Heaviside Function: Clarifying H(x-a) & H(a-x)

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  9. B

    A Integrating definite Heaviside function

    I am trying to integrate the following triple integral, which has a heaviside function in the inner most integral:$$ \frac{16}{c_{4}^{4}} \int_{0}^{c_{4}} c_{3}dc_{3} \int_{c_{3}}^{c_{4}} \frac{dc_{2}}{c_{2}} \int_{0}^{c_{2}}f(x)\left ( 1-H\left ( x-\left ( c_{4}-a \right ) \right ) \right )dx...
  10. S

    A Lorentz invariance of the Heaviside function

    Consider the Heaviside function ##\Theta(k^{0})##. This function is Lorentz invariant if ##\text{sign}\ (k^{0})## is invariant under a Lorentz transformation. I have been told that only orthochronous Lorentz transformations preserve ##\text{sign}\ (k^{0})## under the condition that ##k## is a...
  11. kostoglotov

    An infinite sum of the Heaviside function

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  12. E

    Fourier Transform of Heaviside function

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  13. L

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  14. L

    Retarded Green's Function for D'Alembertian

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  15. R

    Inverse Laplace involving heaviside function

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  16. _N3WTON_

    Laplace Transform of Heaviside Function

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  17. D

    D'Alambertian operator on the heaviside function?

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  18. T

    Integral of Unit step function

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  19. P

    Laplace transform with Heaviside function

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  20. 1

    Unsure about Inverse Laplace Heaviside Function question

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  21. hilbert2

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  22. G

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  23. Kushwoho44

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    Homework Statement Hi guys, so I'm struggling with understanding the heaviside function clearly. In particular, I am struggling to understand how changing f(t) to f(t-3) ''shifts'' everything to the right. Homework Equations Say we take the following heaviside function f(t-3) = 7...
  24. S

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  25. D

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  26. 1

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  27. S

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  28. V

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  29. C

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  30. D

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  31. U

    MHB Solve f(t) with Heaviside Functions: Find b

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  32. P

    Problem with integrating heaviside function.

    Homework Statement I need to find the integral of the following equation.. integral from 0 to 8 of {(2H[x-0]+2H[x-4])*(x/8)}dx The Attempt at a Solution Im am not sure what the integral of the heaviside function is? is it 1 or 0? Any help would be appreciated! Thanks
  33. Z

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  34. A

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    Dear Fellows I need to define a function "f" in MATLAB as f is a heaviside unit step function I did it as f = heaviside[abs(-4)] but it is not working as I required.
  35. K

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  36. C

    Integrate product of heaviside function and function of x

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  37. N

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  38. J

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  39. S

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  40. T

    Expressing a function as Heaviside function

    I have a question: given a piecewise function, f(t)= {t when t<=0 ,0 when 0<t<=1, 1-t when t>1. How do express it as a Heaviside function? I have tried to do the solution which is : f(t)= (1-t)*H(t-1)+t*H(t). Here I have considered two non-zero portions of f(t), which are t and 1-t. Is that...
  41. S

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  42. C

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  43. M

    Heaviside function and Unit impulse (delta) help

    Can someone give me quick refresher on what happens when you multiply the heaviside function with the unit impulse? Typically, the unit step function multiplied by anything simply delays it by the offset in the unit step function. The unit impulse function makes the value defined at only one...
  44. K

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    Homework Statement Hi there, i am trying to do a proof that H'(t)= δ(t) Homework Equations We have been given the following: F is a smooth function such that lim (t-->±∞)F(t)=0 Therefore the integral between ±∞ of [H(t)F(t)]'=[H(t)F(t)]∞-∞=0 I understand it up until this point...
  45. S

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    Homework Statement Find the Laplace of the piecwise continuous function: F(t)= t (when t<2) = 8-3t (when 2<=t<3) = t-4 ( when 3<=t<4) = 0 (when 4<=t) Homework Equations I want to use the heaviside function to see if i can apply it to other questions The Attempt...
  46. R

    Need help regarding real-life application of Heaviside Function

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  47. T

    Heaviside function or singularities function

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  48. S

    Is the square of Heaviside function equal to Heaviside?

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  49. V

    Graph Solution for u(t)-2u(t-2)+u(t-5) with Heaviside Function

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  50. M

    Understanding the Heaviside Function: Solving the Equation for a Graph

    hey guys, I am having a bit of trouble finding the equation for the following graph. I know it should look something like: u(t)-2u(t-2), but I don't really understand it. Can someone point me in the right direction?
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