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Questions tagged [step-function]

A step function, also known as a simple function, is a finite sum of characteristic functions of bounded intervals. They are often used in real analysis and measure theory to approximate integrable functions.

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Riemann integrability for step function

Here is the problem: Fix $c\in\mathbb{R}$ and define $g:[0,2]\to\mathbb{R}$ by $$g(x)=\begin{cases}2 &\text{if } 0\le x<1\\c &\text{if } x=1\\ 1&\text{if } 1< x\le 2.\\\end{cases}$$ ...
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Define this step function over the rational numbers

In desmos I plotted a step function (I only plotted 30% of it). Here is my graph: This function is a function from $\Bbb Q\cap (0,1) \to \Bbb Q\cap(0,1).$ The step function is generated by counting ...
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The product of a Heaviside function and Dirac function centered around different points.

Let $a,b$ be two real-numbers such that $a \neq b$. Let $\iota(x \leq a)$ be the Heaviside step function in variable $x$ and $\delta_b(x)$ be the Dirac Delta function centered around $b$. I have two ...
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Evaluating the integral $\int_0^{1}\int_{0}^{2\pi}x\delta (v-x\cos\theta)dxd\theta$

I want to evaluate the following integral $$\int_0^{1}\int_{0}^{2\pi}x\delta (v-x\cos\theta)dxd\theta$$ According to the book where I found the exercise, the answer is $$C\Theta(1-v^2)\sqrt{1-v^2}$$ ...
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Laplace Transform of a Piece-wise function with a Weibull distribution.

Suppose I have the following piecewise function: $$Q(t) = \begin{cases} W(t) & t<T \\ 1 & t=T \\ 0 & t>T \end{cases}$$ ...
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step function question: What tools can be used to study it?

Consider the step function $$A(x)=\sum_{n=1}^\infty e^{\mathrm{floor}\bigg(\frac{\log n}{\log x}\bigg)+\mathrm{floor}\bigg(\frac{\log n}{\log (1-x)}\bigg)} = \prod_{\mathrm{ p~ prime}} \frac{1}{1-e^{\...
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Hiper Calc app: symbolic integration step functions possible?

The app (android) Hiper Calc is rather powerful, with great CAS-capabilities. But it doesn't seem to have step functions, or Piecewise, or other ways to define conditional functions. I tried to mimic ...
Stef Pillaert's user avatar
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Where did I go wrong in using the residue theorem to find the inverse Laplace transform of this function?

I used the residue theorem to solve the inverse Laplace transform of: $$f(t)=\mathcal L^{-1}\Bigg( {s e^{zs} \over (k-s)^2(k+s)^2}\Bigg) \tag 1$$ where $k$ and $z$ are non-negative. I have two poles ...
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differential equation with delta and heaviside functions

Suppose we have a differential equation given by: $$ \frac{d}{dx}f(x) = g\big(c.H(x)\big)\delta(x) $$ where $H(x)=1_{x\ge0}$ is the Heaviside step function, $c$ is a constant and $g$ is a function (...
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Step functions vs piecewise continiuty

Definition: A function $f :[a, b] \Rightarrow\mathbb{R}$ is said to be piece-wise continuous if there exists a partition $a = t_{0} < t_{1} <...<t_m =b $, such that for each $k$, the ...
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General case of the " integration by parts"

Definition: Given an integrable function $f$ on $[a,b]$ by a primitive of $f$ is meant any function that differs from the function $$F(x):=\int_a^x f$$ by a constant. Definition: A function $g: [a, b]...
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Convolution Integral with Same Direction in Integrand

I am working on the convolution below, however I have gotten stuck. I am not sure how to think about changing the bounds of the integrals to give me an answer. Here is the problem, where $u(t)$ is the ...
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Understanding limits of integration

I have a question on limits of integration. The exercise is as follows: Give an example of a step function f, defined on the closed interval [0,5], which has the ...
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Is the 'Unit step function' indeed a function?

I have read in an article that 'We do not define $u(t)$ at $t=0$. At $t=0$ we think of it as in transition between $0$ and $1$'. The domain of the Unit step function is the real numbers, but if $u(t)$ ...
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2-D inverse Fourier transform of Heaviside function

Now I have a Heaviside function $H(K-\sqrt{k^2+l^2})$ in a 2D $\hat k$ space, where $k$ and $l$ are two variables in that space. In a paper, it is said that the inverse Fourier transform of this ...
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Integration of Heaviside (step) function

Ok, I do not have much experiencing using (or even integrating) Heaviside-step functions, so I am looking for a little help. The integral in question is, \begin{equation} 6m^2\int_0^\infty d\mu \frac{...
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Where is the error in solving this simple ODE with the Dirac delta impuslse and Heaviside step function?

I am trying to solve the following ODE: $$ {dT(\tau) \over d \tau} + \rho ^2 T(\tau)+a H(m\tau-d) + b \delta(m\tau-d)=0 \tag1 $$ where $\rho$, $a$, $b$, $m$, and $d$ are constants, $H$ is the ...
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Problems writing the standard formula of a piecewise function

I have the following plot: which is reflected by the given formula: $$ \chi_{A_j}(\xi)=\begin{cases} 2, \ \ \ \ -2\le \xi<-1 \\ 1, \ \ \ \ -1\le \xi<0 \\ 2, \ \ \ \ ...
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Finding a sequence of step functions which converges uniformly to $x^c$ on $[1,b]$

The question is as follows, Given $b>1$ and $c \in \mathbb{R}$, find a sequence of step functions $f_n$ which converges uniformly to $f(x) = x^c$ on $[1, b]$. Use the partition $1 = b^{0/n} < b^...
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Question on the step-wise function

I have a question regarding the stepwise function properties. I have the stepwise function on the uniform grid: \begin{equation}\label{step} f(t)=\begin{cases} 2, \ \ \ \ -2<t<-1 \\ ...
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What is incorrect in my way for getting Fourier transform of step function?

Today I tried to get Fourier transform of step function ($u(t)$). But I got a result which seems is not correct. I want to know what is incorrect in my work? With ...
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Discrete Fourier Transform of repeating sequences

Based mostly on W. Briggs and V.E. Henson’s “The DFT : An owner’s manual to the Discrete Fourier Transform” chapter 3 and question 57 on rarified and repeated sequences, I would like to find the ...
Firulander Sebalacar's user avatar
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Prove that the integral of a step function is independent of a particular representation [duplicate]

Is the following proof valid? preliminaries: Define a step function as any function that can be written as: $x = \sum_i^m \lambda_i f_i$ where $f_k = 1$ on $[a_k, b_k)$ and $0$ elsewhere (the “...
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If $f:[0,1] \to \mathbb{R}$ is of bounded variation, is $|f'|$ is integrable?

In reading the top-voted answer on this post, the answer appears to use the following fact (in the first bullet point of the answer): Claim: If $f: [0,1] \to \mathbb{R}$ is of bounded variation, then $...
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Easy Lebesgue integral of a non-horizontal line but by definition

Maybe a dumb question based on all the questions I've asked for the last decade, but what's the general way to do the Lebesgue integral of some non-negative (measurable?) function that is Riemann ...
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Exponencial grow, with controlable outcome and steps.

I'm not a mathematician so I'm not sure how to calculate what I need. The problem is, I want to go from 0 to 0.9, growing exponentially, over a determined amount of steps. And with every step added ...
r_ilho's user avatar
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Is the integral of the Dirac delta function equal to the integral of the Dirac delta function times the Heavisde unit step function? [duplicate]

Given that the Dirac delta function is defined as: $$ \delta(t) = \begin{cases} +\infty, & t = 0\\[2ex] 0, & t \neq 0\\[2ex] \end{cases} $$ And that the Heaviside unit step function is defined ...
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Maximum of $y''$ for BVP with $y''''\leq0$.

Consider the following boundary value problem for some $L>0$ and $w(x)\geq 0$: $$\frac{d^4y}{dx^4}=-w(x)\,;\,\,\,y(0)=y(L)=0,\,y'(0)=y'(L)=0.$$ Here $w$ can be pathological: a step-function or an ...
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How to solve a differential equation that has a unit steap as part of it?

Lets say that i have a differential equation $y'(x)+y(x)=x $ the answer will be pretty straightforward right?, $e^{\int(1)*dt}=e^t$, then: $y*e^t=\int(x*e^x)dx $ (we have integration by parts) $y*e^t=...
MastergGM's user avatar
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Algebraic Intersection of a Step Function and another function

I am trying to find an algebraic way to solve for the intersection of a step function and another function (in this case linear but any polynomial function in general). The example is given as follows ...
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Set of step function is dense in piecewise functions set, understanding of the statement.

Note : I think i found the proper mathematical terms that corresponds to what exists in my country , if you notice an obvious error of international term , thanks to correct. Context : Defining ...
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Simplify recursive equation $h(n) = \alpha h(n-1) + (1-\alpha)\delta(n)$

Can anyone tell me how to simplify this recursive equation? $$h(n) = \alpha h(n-1) + (1-\alpha)\delta(n)$$ $\delta(n)$ is Dirac delta function. I have got to this so far: $$h(n) = (1-\alpha)\delta(n) +...
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unit step signal

I have a question that asks to draw the following signal: x[n] = u[n] - u[n-2] I know that δ[n]=u[n]−u[n−1] but we are shifting it by 2 so it's different I don't know how to draw it and how should the ...
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Solving differential equation with step function without using Laplace Transforms.

Suppose we have the differential equation: $$ \ddot{y} + y = H(x - \pi) - H(x - 2\pi) $$ where $ H(x)$ is the Heaviside step function with initial conditions $ y(0) = \dot{y}(0) = 0 $ as initial ...
Grotto Box's user avatar
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1 answer
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What is the antiderivative of the Heaviside step function?

What is the antiderivative of the Heaviside step function $\Theta (x)$ as defined on Wikipedia? I have seen somewhere it is $x\Theta (x)+C$, yet taking the derivative of this expression using the ...
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Antiderivative of Heaviside function with absolute-value-argument

I'm looking to calculate the antiderivative of $$\Theta (R-|x|),$$where $\Theta$ denotes the Heaviside step function and $R$ is a given constant. On Wikipedia it is given that $\int_{-\infty}^x \Theta(...
psie's user avatar
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Showing $\int_{-\infty}^x\delta(a)da = \theta(x)$ for $x\neq0$

I'm trying to show that $\int_{-\infty}^x\delta(a)da= \theta(x)$ for $x\neq0$, where $\delta(x)$ is the dirac delta function, and $\theta(x)$ is the step function , which equal to $0$ for $x\leq0$ and ...
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Continuity at a point vs. interval—contradicton or not?

Let $f(x)=\lfloor x \rfloor $ and imagine posing the following questions. Is $f(x)$ continuous at $x=0$? Is $f(x)$ continuous on $[0,1)$? For the first question, since $\displaystyle \lim_{x\...
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How to propagate error through a Heaviside step function $H(x\pm\epsilon)$

Suppose we have a set of data points $\{x_1,\cdots,x_n\}$ with a corresponding set of errors $\{\epsilon_1,\cdots,\epsilon_n\}$. Using the standard definition of the Heaviside step function, $$H(x)=\...
nebula's user avatar
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Identity to break up a shift in a Heaviside Step Function

$H(x)$ is defined as the Heaviside step function, so $H(x) = \Bigg\lbrace\begin{array}{ll} 1 & x > 0 \\ 0 & x \leq 0 \end{array}$ The case for $H(0) = 0$ matters for me. ...
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Finding the impulse response of a system from its step response

We know the impulse response and the step response have the following relationship: $$h(t) = \frac{dy_{step}}{dt} $$ Given the following step response of a system: $$ y_{step(t)} = \begin{cases} 0, \...
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How can I write a single function expression describing some ellipses in a rectangular domain?

I have a rectangular domain (0<x<a, 0<y<b) and have some filled cylinders with ellipse cross-sections and different sizes which are located separately within this domain. How can I write a ...
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Minimizing a univariate function

Let $r$ and $N$ be positive integers, and let $\epsilon$ be a positive real number for which $2\leq r \leq N-1$ and $\frac{\epsilon}{N-r}\leq \frac{1-\epsilon}{r}$. I would like to analytically solve ...
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Existence and name of "Step Number" scheme for extended derivation of step functions?

Surely the following concept must have been thought of: The extended-derivative of the step function f(x)={x<0 : 0 ; x>=0 : 1 } shall be defined as g(x)={x≠0:0; x==0:(0,1)} Where (0,1) is a &...
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Is my derivation of the derivative of a product of the Heaviside function and a function correct? (A follow-up question)

That's a follow-up question to the accepted answer to this post. After some thinking, I ended up deriving the derivative differently. I'm wondering if the dear Math stack exchange community can tell ...
Ivan Nepomnyashchikh's user avatar
2 votes
3 answers
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Solving $y'' + 2y' + 2y = 2\delta' + 2\delta$ without Laplace transform

Im trying to solve the following differential equation: $$y'' + 2y' + 2y = 2\delta' + 2\delta$$ I did this by first setting $ y(t) = z(t)\theta(t)$ and finding the causal solution to the problem. From ...
maximise_max's user avatar
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2D Interpolation techniques scattered data

I am trying to understand the interpolation technique that someone else has implemented at work. Since I can't ask him, I posted the question here. So I have the following scenario. I am trying to ...
psst_sawpaw's user avatar
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How to prove that $f(t)\delta(t) = f(0) \delta(t)$?

I found this equation in thi s question How to differentiate $f(t)\theta(t)$, the product of a function with the Heaviside unit step? $$ f(t)\delta(t) = f(0) \delta(t) \tag{1} $$ Now I have a similar ...
Zhao Dazhuang's user avatar
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about the derivative of the Heaviside step function

I know that the derivative of the Heaviside step function follows: $${\partial \over {\partial {t_1}}}\theta \left( {{t_1} - {t_2}} \right) = \delta \left( {{t_1} - {t_2}} \right)$$ what about if the ...
Bekaso's user avatar
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2 answers
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Floor inequality in proof

I am currently trying to understand the proof that $[-x] = -[x]-1$ if $x$ is not an integer (solutions for $b$.) where $[x]$ is the floor function. Can somebody explain me how he went from $$-n-1 < ...
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