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. F

    Charge invariance with Heaviside's function

    I followed a demonstration in one of my electromagnetism books, but it is not clear to me. My problem is at the starting point. The book begins by considering the office defined in the following way: $$Q=\int d^4xJ^\alpha(x)\partial_\alpha\theta(\eta_\beta x^\beta)$$ where...
  3. B

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

    Homework Statement:: Why is the heaviside function in the inverse laplace transform of 1? Relevant Equations:: N/A This is a small segment of a larger problem I've been working on, and in my book it gives the transform of 1 as 1/s and vice versa. But as I've looked online for help in figuring...
  4. 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...
  5. C

    Converting a Piecewise function to a Heaviside function

    Homework Statement Use the Heaviside function as an on the switch over the interval [a,b]. Homework Equations Let the H(x) be the Heaviside function defined as a piece-wise function such that it is zero if x is less than zero, and 1 if it is greater than or equal zero. From that, we can use the...
  6. B

    I Heavyside step function chain rule

    Hi, I have a probably very stupid question: Suppose that there is an expression of the form $$\frac{d}{da}ln(f(ax))$$ with domain in the positive reals and real parameter a. Now subtract a fraction ##\alpha## of f(ax) in an interval within the interval ##[ x_1, x_2 ]##, i.e. $$f(ax)...
  7. Sudhir Regmi

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

    Hi everyone, I could not figure out whether H(x-a) and H(a-x) are same or different. Please help me to understand this.
  8. 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...
  9. 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...
  10. kostoglotov

    An infinite sum of the Heaviside function

    I'm not sure where to put this question. It is by itself pretty basic, but it's a preamble to a Laplace Transform exercise, and I'll probably want to ask some follow up questions once the current query is resolved. 1. Homework Statement Unit stair-case function: f(t) = n, \ if \ \ n-1 \leq t...
  11. E

    Fourier Transform of Heaviside function

    Homework Statement Find the Fourier transform of H(x-a)e^{-bx}, where H(x) is the Heaviside function. Homework Equations \mathcal{F}[f(t)]=\frac{1}{2 \pi} \int_{- \infty}^{\infty} f(t) \cdot e^{-i \omega t} dt Convolution theory equations that might be relevant: \mathcal{F}[f(t) \cdot...
  12. L

    Simple (Constant) Wavefunction -- Find Uncertainty In p^2

    Homework Statement Given the following wave function valid over -a \le x \le a and which is 0 elsewhere, \psi(x) = 1/\sqrt{2a} Find the uncertainty in \left<\left(\Delta p\right)^2\right> momentum, and the uncertainty product \left<\left(\Delta x\right)^2\right>\left<\left(\Delta...
  13. L

    Retarded Green's Function for D'Alembertian

    Hey All, I recently posted this in another area but was suggested to put it here instead. Here is my original post:
  14. R

    Inverse Laplace involving heaviside function

    Homework Statement Give the inverse Laplace transform of F(s) = (-3/s) + (e^-4s)/(s^2) + (3e^-4s)/s Homework Equations Inverse Laplace [e^(-cs) F(s)] = f(x-c)u(x-c) The Attempt at a Solution I'll break this into 3 parts. Part 1 - (-3/s) -3/s = -3(1/s) -> inverse laplace of -3(1/s) = -3...
  15. _N3WTON_

    Laplace Transform of Heaviside Function

    Homework Statement Determine the Laplace transform of the given function: f(t) = sin(t) for 0 <= t < \pi and f(t) = 0 for \pi <= t Homework EquationsThe Attempt at a Solution Ok, I've been having some trouble figuring out how I should write the above branched function (sorry for the...
  16. D

    D'Alambertian operator on the heaviside function?

    Hey guys, How does one compute the following quantity: \Box \theta(x_{0})=\partial_{0}\partial^{0}\theta(x_{0})? I know that \partial_{0}\theta(x_{0})=\delta(x_{0}) which is the Dirac delta, but what about the second derivative? Thanks everyone!
  17. T

    Integral of Unit step function

    This isn't really so much of a specific example from a textbook, I just need to understand how to do this kind of integral and from that I can infer how to do all of the other problems. 1. Homework Statement The current problem I'm having trouble with is: Integral of (5u(t-1)) from T to -T...
  18. P

    Laplace transform with Heaviside function

    Hello, I am searching for the Laplace transform of this function u_a(y)\frac{\partial c(t)}{\partial t} where u_a(y) is the Heaviside step function (a>0). Can anyone help me? Thanks in advance! Paolo
  19. 1

    Unsure about Inverse Laplace Heaviside Function question

    Homework Statement Find the inverse Laplace transform of F(s)=5e^(-8s)/(s2+36) Homework Equations The Attempt at a Solution I know that to find the inverse Laplace transform of this function, I start by factoring out (e^(-8s)) to end up with 5/(s^2+36), and that my final answer...
  20. hilbert2

    Rational approximation of Heaviside function

    Hi, could someone please help me with this one: I'd need to form a sequence of rational functions ##R_{n}(x)## such that ##\lim_{n \to \infty} R_{n}(x)=\theta(x)##, where ##\theta(x)## is the Heaviside step function. The functions ##R_{n}(x)## should preferably be limited in range, i.e. for some...
  21. G

    Laplace transform of Heaviside function

    Homework Statement What is the laplace transform of H(-t-17) Homework Equations Shifting theorem: L(H(t-a)) = (e^-as)/s The Attempt at a Solution This is the only part of the problem that I can not get (this part is from a larger differential equation I'm trying to solve). I'm can't seem...
  22. Kushwoho44

    Conceptual Misunderstanding in dealing with Heaviside Function

    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...
  23. S

    Evaluating integral involving Heaviside function.

    Homework Statement Evaluate ∫ (t - 1)^2 U(t - 2)dt on the interval [0, 5] Homework Equations τ = 2, b = 5. U(t - τ) = 0, t < τ and 1, t > τ. The Attempt at a Solution Decompose integral up into two parts [0, 2] and [2,5]. U(t - 2) will = 0 on the first interval as t < τ and it...
  24. D

    Partial differential equation-delta Dirac& Heaviside function

    I got 2 questions to ask! I have finished one but not sure if it's correct so I need to double check with someone :) http://imageshack.us/a/img708/1324/83u8.png Here is my worked solution, I took this picture with my S4 and I wrote is very neatly as I could! The reason I didn't type it all...
  25. 1

    Differential Equation Involving Heaviside function

    Homework Statement y''+y= H(3pi)(t) y(0)=1 y'(0)=0 H(3pi)(t) means the heaviside function such that it is 1 above 3pi and 0 otherwise. Homework Equations The Attempt at a Solution Taking the laplace of both sides, I have that: L(y) = (L(H(3pi)(t))+s) / (s^2 + 1)...
  26. S

    Complex vector potential of solid sphere and Heaviside function

    Homework Statement I have done this problem for the case of a spherical shell, however, I am not understanding how to go about this for a solid sphere. Homework Equations \vec{A} = \frac{1}{4 \pi} \int_{\phi' = 0}^{2 \pi} \int_{-1}^1 \int_0^R \rho_o \Theta(R-r) \sum_{l=0}^\infty...
  27. V

    Laplace transform heaviside function

    I have a question asking for the inverse laplace transform of (e^(-s))/(s^2+pi^2). I split it up to (e^(-s))/s x s/(s^2+pi^2) and got u(t-1)cos(pi(t-1)),but the correct answer is (sin(pi(t-1)/pi)u(t-1). So here it was split up to (e^(-s))/pi x pi/(s^2+pi^2) and I don't understand where the...
  28. C

    Second Derivative of Heaviside Function

    Hello all. In short, I am wondering what the second derivative of the Heaviside function (let's say H[(0)]) would be. I'm presuming that it's undefined (or more accurately, zero everywhere but at x=0), but I would like to know if that is correct. Essentially, I am attempting to extend a...
  29. D

    MHB Continuous Except Jump Discontinuity: Showing True for Any Function

    Show that a function on (a,b) that is continuous except for a jump discontinuity at $x_0\in(a,b)$ is of the form $$ f(x) = g(x) + cH(x - x_0) $$ where c is a constant and g is continuous on (a,b) except possibly for a removable discontinuity at $x_0$. I know that is true since this how I...
  30. U

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

    Consider the function Then f(t) can be expressed with Heaviside functions as Determine b. Workdone so far... f(t) = 7H(t) + bH(t-5) when t = 5 f(5) = 49 + 7b 15 = 49 + 7b -34 = 7b b = -34/7 http://www.chegg.com/homework-help/questions-and-answers/-q3136934# Is the answer correct...
  31. 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
  32. Z

    What is Heaviside Function? Tutorial & Application

    Hi all: The Heaviside function multiples a random varaible, is that a probability density function? This is my first time knowing about Heaviside, any tutorial and application of it?
  33. A

    MATLAB What is the syntax for defining a Heaviside function in MATLAB?

    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.
  34. K

    Fourier Transform of exponential and heaviside function

    Homework Statement Compute the Fourier transform of \phi(t)=(e^(-at))H(t) where H(t) is the Heaviside step function Homework Equations The Attempt at a Solution I am stuck in an attempt at the solution, I am confused at how the heaviside step function factors in and think...
  35. C

    Integrate product of heaviside function and function of x

    How would you integrate product of heaviside function and function of x i.e. int[f(x)H(x)dx] Thanks
  36. N

    How do you do a gaussian integral when it contains a heaviside function?

    How do you do a gaussian integral when it contains a heaviside function!? Very few textbooks cover gaussian integrals effectively. This isn't a big deal as they are easy to locate in integral tables, but something I cannot find anywhere is how to handle a gaussian with a heaviside heaviside...
  37. J

    Sum to infinity of Heaviside function

    I'm revising my course text for my exam and came across a Fourier series problem finding the Fourier series of the square wave: http://img574.imageshack.us/img574/5862/eq1.png. It is then calculated that the complex Fourier coefficients are...
  38. S

    Heaviside Function: What is H(-x)? Fourier Transform

    What is H(-x)? Is it 1 if x < 0, 0 if x \geq 0. If so what is the Fourier transform of H(-x)exp(x)?
  39. 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...
  40. S

    The Heaviside function Help

    Homework Statement Find the solution for L[y]=H(t-pi/2)sint=q(t) y(0)=0 by using the Green's function. Homework Equations The Attempt at a Solution My problem is that I'm stuck with how to get the Heaviside into the required general solution. Is it anything to do with the fact that the...
  41. C

    Heaviside Function Homework: Laplace Transform w/ Right Shift

    Homework Statement how to i take the laplace transform of this , -tH(t-1) so we need to get thr right shift so is it -(t-1 + 1 ) so do i take the laplace transform of -(t+1) so would it be -(1/s^2 + 1/s ) *e^(-s)
  42. 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...
  43. K

    Heaviside function and dirac delta

    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...
  44. S

    Laplace (heaviside function)

    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...
  45. R

    Need help regarding real-life application of Heaviside Function

    Hello. Can someone tell me how is Heaviside (or Unit Step) Function used in real life?
  46. T

    Heaviside function or singularities function

    Does anybody knows heaviside function or singularities function such as <x-a>^n ;n=1,2,3... for mechanics i wish to know what does this mean and how to draw the shears stress and moments diagramusing this function.any recommended websites for this is appreaciable too thanx
  47. S

    Is the square of Heaviside function equal to Heaviside?

    Is the square of Heaviside function equal to Heaviside? H(t-t')\times H(t-t')=H(t-t')? Please help me on the above equation.
  48. V

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

    can someone please help me graph this function: u(t)-2u(t-2)+u(t-5) where u(t) is heaviside fcn. i got that it's 1 from 0<t<2 and 1>5 but i don't think that's right but don't know what to do with the 2 in 2u(t-2). thank you.
  49. 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?