# What is Step function: Definition and 142 Discussions

In mathematics, a function on the real numbers is called a step function (or staircase function) if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.

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1. ### Antiderivative of Heaviside step function with absolute-value-argument

For ##R<0##, the antiderivative is just a constant, since then ##R-|x|## is negative for all values of ##x##, which in turn implies ##\Theta(R-|x|)## is zero for all values of ##x##. For ##R\geq 0##, and by inspection apparently, the antiderivative is ##(R+x)\Theta(R-|x|)+2R\Theta(x-R)+C.##...

12. ### I How to invert unit step function?

I am trying to find out how to reverse the unit step function. The closest I could find is this sentence, which is more like a definition? "if we want to reverse the unit step function, we can flip it around the y-axis as such: u(-t). With a little bit of manipulation, we can come to an...
13. ### Efficient Integration of Step Function with Variable Denominator

The problem I want to calculate ## \int^6_{-6} \frac{g(x)}{2+g(x)} \ dx ## for the step function below.The attempt I started with rewriting the function as with the help of long-division ## \int^6_{-6} \frac{g(x)}{2+g(x)} \ dx = \int^6_{-6} 1 \ dx - 2\int^6_{-6} \frac{1}{g(x)+2} \ dx## I know...
14. ### Step function Laplace transform - can someone check my work?

Homework Statement Homework EquationsThe Attempt at a Solution 1 ##3u(t)-3u(t-2)## ##\frac{3}{s}-\frac{e^{-2s}3}{s}## 2 ##\frac{5t}{2}u(t)-\frac{5t}{2}u(t-2)## ##\frac{5}{2s^2}-\frac{5e^{-2s}}{2s^2}## 3 I have no idea how to do this one.
15. ### I What happens in this step function?

when the x>0,then the theta is equal to 1 So the theta =0 when x=0 in the second term of the integral, it starts to integrate the function from "0 "to infinity(see the yellow loop inside)Since the" 0 "should not be included ,otherwise theta(0)=0 and (0*df/dt)=0 but why the 0 is still...
16. ### A Integration using delta function and step function

I would like to evaluate the following integral: ##\displaystyle{\int_{-\infty}^{\infty} dp^{0}\ \delta(p^{2}-m^{2})\ \theta(p^{0})}## ##\displaystyle{= \int_{-\infty}^{\infty} dp^{0}\ \delta[(p^{0})^{2}-\omega^{2}]\ \theta(p^{0})}## ##\displaystyle{= \int_{-\infty}^{\infty} dp^{0}\...
17. ### Laplace Transform unit step function

Homework Statement g(t) = (t+1)us(t) - (t-1)us(t-1) - 2us(t-1) - (t - 2)us(t-2) + (t-3)us)(t - 3) + us(t-3) Homework Equations unit step function us(t-3) is same as u3 (t) Shift in time: L[f(t - T)us(t-T)] = e-TsF(s) us(t) ↔ 1/s t ⇔ 1/s2 The Attempt at a Solution 1/s2 + 1/s - e-s/s2 + e-s/s -...
18. ### Derivative of unit step function

Homework Statement Show that δ(x-x') = d/dx Θ(x-x') Homework Equations ∫ f(x') δ(x-x') dx' = f(x) Θ(x-x') vanishes if x-x' is negative and 1 if x-x' is positive The Attempt at a Solution I saw a relation of the δ function but I don't know why is it like that. Integral of δ(x-x') from -∞ to x...
19. ### Strange integral of heaviside step function

I ran across this integral while reading Ashcroft and Mermin's solid state physics book... ∫Θ(f - f(t) )dt = t_max - t_min Where Θ is the heaviside step function and the integral runs from 0 to infinity. Does anyone have any idea how this integral makes sense?
20. ### Step Function Laplace w/ piecewise

Homework Statement ##y'' + 2y' + 2y = h(t); y(0)=0, y'(0)=1, h(t) = \begin{cases} t &, \pi \leq t < 2\pi \\ 0 &, 3\pi \leq t < \infty \\ \end{cases}## 2. Homework Equations The Attempt at a Solution Take the Laplace of both sides:[/B] ##\mathcal{L}(y'' + 2y' + 2y) = \mathcal{L}(h(t))...
21. ### Step Function IVP Differential Equation w/ Laplace Transform

Homework Statement (didn't know how to make piecewise function so I took screenshot) Homework EquationsThe Attempt at a Solution My issue here with this problem is that I have absolutely no idea where to start... I have read through the textbook numerous times, and searched all over the...
22. ### Definite integral of step function

I need to prove whether this expression is true or false: ## \sum\limits_{k=1}^{n}\int_{k-1}^{k}[x]dx = \frac{n(n-1)}{2} ## I'm so confused because as I know, definite integral is possible only when the target function is continuous in closed interval. In this case, function ##[x]## should be...
23. ### Laplace Transform question with unit step function

Homework Statement Homework Equations Laplace Trasformations The Attempt at a Solution a. done b. f(t)= t -3*t*u(t-1) + 4*u(t-1) -3*u(t-2) -2*t*(t-2) c. 1/(s^2) - (3e^-s -2e^-2s)/(s^3) + (4e^-s -3e^-2s)/s d. 1/(s-1) * (1/(s^2) - (3e^-s -2e^-2s)/(s^3) + (4e^-s -3e^-2s)/s) These are the...
24. ### MATLAB Help: MATLAB time step function

Hi guys I'm new to MATLAB but now i need to use it for my study. I've tried to learn by myself to run it but unfortunately it failed, so I need some guidance on how to run following time step function in MATLAB St = St-1 + Pt - ETt, St-1 + Pt - ETt ≤ Smax = Smax, St-1 + Pt - ETt > Smax then...
25. ### Fourier Series of a step function

Homework Statement [/B] f(x)=\left\{\begin{array}{cc}0,&\mbox{ if } 0< x < 2\\1, & \mbox{ if } 2<x<4\end{array}\right. Show that the Cosine Fourier Series of f(x) for the range [0,4] is given by: A + B\sum^{\infty}_{n=0}\frac{(-1)^n}{(2m+1)}cos(\frac{(2m +1) \pi x}{2}) Homework Equations...
26. ### Finding Solutions to a Step Function Integral

Homework Statement This is from Apostol's Calculus Vol. 1. Exercise 1.15, problem 6.(c) Find all x>0 for which the integral of [t]2 dt from 0 to x = 2(x-1) Homework Equations [t] represents the greatest integer function of t. The Attempt at a Solution [/B] Integral of [t]2 dt from 0 to x...
27. ### Fourier Transform of product of heaviside step function and another function

Homework Statement A damped harmonic oscillator is driven by a force of the form f(t)=h(t) t^2 Exp(-t), where h(t) is a Heaviside step function. The Oscillator satisfies the equation x''+2x'+4x=f(t). Use pencil-and-paper methods involving Fourier transforms and inverse transforms to find the...
28. ### Help me to survive from a big dilemma(Unit step function)

Consider: u(t)=\begin{cases} 1\quad \quad \quad \quad t>0 \\ 0\quad \quad \quad \quad t<0 \end{cases} Now I want to calculate this: \int _{ 0 }^{ a }{ \frac { u(t)-u(t-a) }{ { t }^{ 2 } } } dt whereas: a>0 What is confusing me is this point that should our answer for the integral include...
29. ### 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...
30. ### Decomposing Heavy Side Step Function into even and odd components

Homework Statement Decompose H(x) into even and odd components Homework Equations e(x) = (f(x)+f(-x))/2 o(x) = (f(x)-f(-x))/2 The Attempt at a Solution i said f(x)=1 and f(-x)=0 and I got e(x)=1/2 and o(x)=1/2. but this isn't true. e(x)=1/2 but o(x) = 1/2 when x>0 and -1/2 when x<0. what...
31. ### Evaluating and integral including unit step function

how can this integral be calculated: ∫[e^(−2mx) θ^2(x)+2θ(x)θ(−x)+e^(−2mx)θ^2(−x)]dx from -∞ to ∞ where θ(x) is the unit step function with its amplitude 0 everywhere before x=0 and θ(−x) is the unit step function with its amplitude 0 everywhere after x=0In Introduction to Quantum Mechanics...
32. ### Laplace Transform for shifted Unit Step Function

Hello, I have a relatively simple question. after being unable to find it through google I have decided to ask you guys if you know what the Laplace transform of a unit step function that looks like this would look like Us(t-2) From tables, the Laplace transform for a regular units step...
33. ### Integral containing a delta function and a step function

Homework Statement (a) Show that that δ(a-b)=∫δ(x-a)δ(x-b)dx (b) Show that ∂/∂x θ(x) = δ(x) where θ(x) is the heaviside step function (0 for x<0, 1 for x>0) (c) Show that ∫(-inf to inf) δ(x) f(θ(x))dx=∫(0 to 1) f(y)dy Homework Equations The definition of the delta function...
34. ### Laplace transform of unit step function

Homework Statement The Attempt at a Solution I know that u(t) is a unit step function and holds a value of either 0 or 1. In laplace transform, when we integrate f(t) from 0 to infinity, we take u(t) to be 1. In this case, since u(t) is u(-t), does this mean it holds a value of 0? Does not...
35. ### 1st order linear DE with step function input

hi guys, this is my first post, but I've read many. so my problem is actually for an engineering class, but it's more math/physics related. i took DE last semester and i know how to solve linear eqs but the step function/force is throwing me for a loop. 1. Homework Statement we are given a...
36. ### MHB Solve Unit Step Function Algebraically - Help Needed

How would I go about algebraically (not graphically) performing operations on two or more Heaviside function.H(x) = {0, if x<0 ...{1, if x>=0 ex. define the function piecewise and graph. a.) (x+1)*H(x+1)-x*H(x) b.) (x+1)*H(x+1) i'm having a hard time solving this please help me!
37. ### How do I plot this periodic step function with GNUplot?

\sum_{k=0}^{∞} (t-2k) [u(t-2k)-u(t-2(k+1))] = f(t) where u is the step function and the graph of this is supposed to be 45 degree lines repeating to infinity. Sort of like / / / / / / / / / ad infinitum. I took this equation out of this lecture note on page 10. Fig 5.4 is supposedly the graph...
38. ### Finding impedance that is unit step function

Homework Statement Consider the following circuit which uses ideal components. Prior to t=0 switch S is open. Then suddenly at t=0 switch S is closed. Find the impedance Z_{2} such that the system output is a unit step function of voltage. Be certain to show all components used to construct...
39. ### FIR filters, impulse responses and unit step function

Hey everyone, the question I am faced with is this: Which of the following expressions involving δ[n] is incorrect? where "m" is a non zero integer and u[n] is the unit step function. A. u[n-m] = δ[n] + u[n-m+1] B. x[n]δ[n-m] = x[n-m] C. δ[n] = u[n] - u[n-1] D. δ[n]δ[n-m] = 0...
40. ### Differential Equation with unit step function and Laplace transforms

EDIT: Nevermind I see what I did wrong near the end. Homework Statement x'' + 4x = f(t) Where f(t) is 1 if t is between 0 and π, 0 if t > π. Initial conditions are x(0) = x'(0) = 0. Homework Equations Transform of a derivative: L(f^{(n)}(t)) = s^nF(s) - s^{n-1}f(0) -...-f^{n-1}(0)...
41. ### How to find La Place transform of cos(x) * unit step function (x - pi)

Homework Statement Find the La Place transform of cos(x)*(u(x-\pi)) Homework Equations L{u(t-a)}(s)=(e^(-as))/s The Attempt at a Solution I don't think I can just multiply this by the La Place transform of cos (x), which is s/(s^2) ?
42. ### Is x(t) equal to the shifted step function u(t+2)-u(t)?

When you shift step function u(t-a), is it true for a<0? Im sorry but we did only exercises when you don't have to use this and in my homework using this is the only way i can solve it. Like in this pic. Isnt x(t) (forgot to mark the axis, sorry) x(t)=u(t+2)-u(t)...
43. ### Can the DTFT of a Unit Step Function be Simplified for u(n) - u(n-L)?

So I'm trying to find the DTFT of the following; where u(n) is the unit step function. u \left( n \right) =\cases{0&$n<0$\cr 1&$0\leq n$\cr} I want to find the DTFT of u \left( n \right) -2\,u \left( n-8 \right) +u \left( n-16 \right) Which ends up being a piecewise defined function...
44. ### Derivative of a unit step function

So I'm busy struggling with some worked examples in my signals class. I understand the theory from the notes and textbook but I cannot seem to apply them to proper examples. We are asked to find the derivative of g(t) = (1-e^(-t))*u(t) where u(t) is a unit step function. I know the...
45. ### Simple step function, Laplace transform

Homework Statement A system is characterized by the equation y' + 3y = r' . When the input is r(t) = u(t) - u(t-1), find y(t) by taking the inverse Laplace transform of Y(s).Homework Equations The Laplace transform integral The Laplace transform of a derivative sF(s) - f(0) The transfer...
46. ### Integral of delta function multiplied by a Heaviside step function

Homework Statement consider two functions:ψ(x) which is eqaul to zero at a,that is ψ(a)=0 and f(x)=H(x-a)*β(x)+(1-H(x-a))*γ(x) where H(x-a) is the heaviside step function and β(x),γ(x) is the continuous function. it seems that the derivative of f(x) is not exist. the question is whether...
47. ### Is my unit step function correct for solving this differential equation?

Homework Statement y''-4y'-32y={1 when 0<=t<1 and 0 when 1<=t y(0)=y'(0)=0 Homework Equations The Attempt at a Solution s2L(y) -4sL(y)-32 L(y)=u1(t) I am just struggling to figure out if my unit step function is correct. Solving for L(y) I get: (e-s) / (s(s2 -4s-32))...
48. ### Grad 1D scattering from step function

Homework Statement Consider the 1D potential V(x) such that V(x)=0 for x<0 and V(x) = V for x>0 and assume that a wave packet with energy E0=p20/2m<V is incident on the barrier from the left. Calculate in terms of E0 and V the difference in time between the arrival of the incident packet at...
49. ### Inverse Laplace transform with unit step function

Hello again. First off, I wasn't sure how to say this in the title but I'm not taking the inverse Laplace transform of a unit step function. I'm taking the Laplace transform of something that comes out to the unit step function. I have this question, which is a similar version of the...
50. ### Laplace Transform of Step Function

Homework Statement Solve y'' + y = f(t), y(0)=0, y'(0)=1, f(t)= (0 for 0<t<pi) (1 for pi<t<2pi) (0 for t>2pi) The Attempt at a Solution y'' + y = upi(t)-u2pi(t) s2L{y} -sy(0) -y'(0) +L{y} = L{upi(t)} -Lu2pi(t)} L{y}(s2+1) -1 = (e-pi*s/s) -(e-2pi*s/s) L{y} =...