Step function Definition and 136 Threads

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.##...

Non-homegenous first order ODE so start with an integrating factor ##\mu## $$\mu=\textrm{exp}\left(\int a dt\right)=e^t.$$ Then rewrite the original equation as $$\frac{d}{dt}\mu y = \mu g(t).$$ Using definite integrals and splitting the integration across the two cases, $$\begin{align}...

In QFT by peskin scroeder page 30 the action of Klein Gordon Operator on propagator (∂2+m2)DR(x-y)=∂2θ(x0-y0)... how to compute this ∂2θ(x0-y0)?

The answer in the textbook writes: $$ f(x) = \frac{1}{4} +\frac{1}{\pi}(\frac{\cos(x)}{1}-\frac{\cos(3x)}{3}+\frac{\cos(5x)}{5} \dots) + \frac{1}{\pi}(\frac{\sin(x)}{1}-\frac{2\sin(2x)}{2}+\frac{\sin(3x)}{3} + \frac{\sin(5x)}{5}\dots)$$ I am ok with the two trigonometric series in the answer...

Here's my attempt: I would like to confirm my answers.

Honestly not sure how to go about this. Again this is one equation of 4 that I have. I considered using Laplace transforms but taking the Laplace transform of a step function whose argument is one of the variables being solved for doesn't seem possible. Also, if there is an alternative way to...

i could use a bit of tutelage with Matlab. I have a rather simple equation I would like to plot. I want to create a rational series of primes divided by their corresponding W value from the equation I have. P are primes 2,3,5,7,11,13... I am still working on this. Thanks

I wrote some code that is supposed to give me the step function: function y=step_fn(x,a,b) %This is the step function, given a range from a to b, it wil produce a %function which is 1 from a to b and zero otherwise. L=length(x); y=zeros(1,L); n=find(x<=a,1,'last'); if (b==max(x)) m=L; else...

Homework Statement A beam of particles of mass m and energy E is incident from the right unto a square well potential given by ##V(x)=-V_0## for ##-a<x<0##, and ##V(x)=0## otherwise. Solve the Schrodinger equation to determine the wave function which describes this situation. Determine the...

Hello, can we make a Fourier series expansion of a (increasing or decreasing) step function ? like the one that I attached here. I just want to know the idea of that if it is possible.

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)...

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...

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...

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.

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...

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}\...

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 -...

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...

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?

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))...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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!

\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...

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...

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...

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) ?

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)...

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...

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...

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...

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))...

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...

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} =...

Homework Statement Find laplace trasform of g(t) g(t) = 0 for 0<t<1, t^2 for t>1 So this can be re-written as g(t) = t2*u1(t), where u is the unit step function. By using the fact that L(uc(t)f(t-c)) = e-csF(s) i am trying to take the laplace... So in this case, f(t-c) = t2 ... so then does...

Homework Statement I need to solve the ODE y''-y = t - tH(t-1); y(0)=y'(0)=0 Homework Equations - The Attempt at a Solution I'm fine with the process of solving the ODE, but I need a little help regarding the first t. From what I understand from lectures, all of the 't' terms need to be in...

I am trying to see if I understand the convolution process correctly. This is from the solved example of my prof's notes: x[n] = u[n] and h[n] = a-nu[-n] for 0<a<1 As expected the first step was y[n] = x[n]* h[n] = Ʃ h[k]x[n-k] -∞<k<∞ = Ʃ a-ku[-k]u[n-k] -∞<k<∞ From my...