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.
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,
$$\begin{align}...
The answer in the textbook writes: $$ 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)$$
I am ok with the two trigonometric series in the answer...
Honestly not sure how to go about this. Again this is one equation of 4 that I have. I considered using Laplace transforms but taking the Laplace transform of a step function whose argument is one of the variables being solved for doesn't seem possible. Also, if there is an alternative way to...
i could use a bit of tutelage with Matlab. I have a rather simple equation I would like to plot. I want to create a rational series of primes divided by their corresponding W value from the equation I have. P are primes 2,3,5,7,11,13... I am still working on this. Thanks
I wrote some code that is supposed to give me the step function:
function y=step_fn(x,a,b)
%This is the step function, given a range from a to b, it wil produce a
%function which is 1 from a to b and zero otherwise.
L=length(x);
y=zeros(1,L);
n=find(x<=a,1,'last');
if (b==max(x))
m=L;
else...
Homework Statement
A beam of particles of mass m and energy E is incident from the right unto a square well potential given by ##V(x)=-V_0## for ##-a<x<0##, and ##V(x)=0## otherwise.
Solve the Schrodinger equation to determine the wave function which describes this situation. Determine the...
Hello, can we make a Fourier series expansion of a (increasing or decreasing) step function ? like the one that I attached here. I just want to know the idea of that if it is possible.
Hi,
I have a probably very stupid question:
Suppose that there is an expression of the form $$\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
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
(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...
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)...
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
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...
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))...
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...
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...