Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English mathematician and physicist who brought complex numbers to circuit analysis, invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vector calculus, and rewrote Maxwell's equations in the form commonly used today. He significantly shaped the way Maxwell's equations are understood and applied in the decades following Maxwell's death. His formulation of the telegrapher's equations became commercially important during his own lifetime, after their significance went unremarked for a long while, as few others were versed at the time in his novel methodology. Although at odds with the scientific establishment for most of his life, Heaviside changed the face of telecommunications, mathematics, and science.
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.##...
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...
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...
So, Heavside's method confused me, I mean, but we can't divide by zero, can we?
So, my mind sees it as cheating, and to multiply both sides by zero to cancel zeros, and for the example above(from Thomas Calulus), when x = 1, before you multiply by x-1(=0), you get: 2/0=A/0+B/-1+c/-2, and I think...
Richard Feynmann, in His lectures on Physics, Vol I, chapter 28. Gives an expression for calculating the E and B field of a moving point charge based on retarded radius; eg: Knowledge of where the charge was at the time when the field was being formed. The Equation was originally derived by...
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...
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...
Homework Statement
I am trying to show that ## \int \delta (x-a) \delta (x-c) dx = \delta (-a-c) ## via integeration by parts, but instead I am getting ##\delta (c-a) ## (or ##\delta (a-c)## depending how I go...).
Can someone please help me out where I've gone wrong: struggling to spot it...
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...
1.There is an electric field between ionosphere and ground, also Earth's magnetic field is perpendicular to this E field, hence by poynting theorem integrating E x H across the entire surface of Earth gives a great deal of energy flowing west to east.
2. |S| = (E X B)/u0 = (150 V/m X...
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...
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...
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...
Homework Statement
[/B]
I know for t[u(t)-u(t-2)], we can simplify that to tu(t)-((t-2)+2)u(t-2) which then gives us tu(t)-(t-2)u(t-2)-u(t-2). Now, the laplace transform seems trivial but I am having problems with this equation:
sin(t)[u(t)-u(t-pi)]
Homework...
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
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...
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...
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...
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!
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
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...
Homework Statement
Show that the explicitly covariant expression:
GR(x-y) = θ(x0-y0)δ((\vec{x}-\vec{y})2)/2\pi
agrees with the retarded Green function:
δ(x0-y0-|\vec{x}-\vec{y}|) / (4\pi|\vec{x}-\vec{y}|)
Homework Equations
N/A
The Attempt at a Solution
I know that the...
I am trying to solve the integral
\int_{-\infty}^\infty H(x) \delta(x) dx
Where H(x) is a unit step and d(x) is a standard Dirac delta. Mathematica chokes on this, but I'm pretty sure that the value is
\int_{-\infty}^\infty H(x) \delta(x) dx = \dfrac12 \left(H(0^+) + H(0^-) \right) =...
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...
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...
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...
Hi,
If you have a Heaviside function such as (u(x-2)-u(x-6), then am I right in thinking that this would be = 1 or 'on' between x=2 and x=6? What happens if you put x in front of the equation? E.g. x(u(x-2)-u(x-6))
Im trying to understand Heaviside but I am getting quite confused by it...
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...
Good day! I would just like to ask on how to convert a particular piece wise function to a heaviside equation.
For a function,
f(t) = t2 if t < 2
t - 1 if t >= 2
the heaviside equation would be,
H(t) = t + (t-1 -t)u2(t)
or
H(t) = t - u2(t)
but i can't comprehend on...
Homework Statement
Plot this signal
y(t)=3u(t+3)-r(t+2)+2r(t)-2u(t-2)-r(t-3)-2u(t-4)
The Attempt at a Solution
I tried to draw each single function, but do not know the way to plot for the whole signal like the bottom graph
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...
I'm working on a basic potential step problem from Griffiths QM. Its problem 2.34 in the textbook. I am having trouble understanding the transmission coefficient in this problem. I realize that I can calculate the reflection coefficient and just do 1 minus the result but I am looking for...
Homework Statement
(6-t)heaviside(t-2)
This is just one term of the real problem I'm working, but it will serve to help me figure this out.
Homework Equations
The Attempt at a Solution
http://www.wolframalpha.com/input/?i=laplace+transform+%7B%286-t%29heaviside%28t-2%29%7D...
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)...
Homework Statement
Rearrange f(t) using Heaviside Step Function
Then Rearrange it so that the Laplace Transform can be written down
Then, Write the Laplace Transform of f(t)
Homework Equations
The Attempt at a Solution
So my first step is as follows...
Using the basic Piecewise Function in...
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
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...
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...
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...
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...
Homework Statement
∫(H(t-2)t)dt from t=0 to t.
Homework Equations
The Attempt at a Solution
It's same as ∫t=2 t (t-2)dt, which is
t^2/2-2t+2.
But do we have to multiply the answer with H(t-2)?
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...
Homework Statement
Find the solution of the equation:
α(dy/dt) + y = f(t)
for the following conditions:
(a) when f(t) = H(t) where H(t) is the Heaviside step function
(b) when f(t) = δ(t) where δ(t) is the delta function
(c) when f(t) = β^(-1)e^(t/β)H(t) with β<α
Homework...
Find the solution of the equation:
α(dy/dt) + y = f(t)
for the following conditions:
(a) when f(t) = H(t) where H(t) is the Heaviside step function
(b) when f(t) = δ(t) where δ(t) is the delta function
(c) when f(t) = β^(-1)e^(t/β)H(t) with β<α
My try for all 3 are as follow:
1...
Good evening everyone,
I have a final exam where one of the questions will most likely be a heaviside function and using Laplase transforms since we just studied it. I am having trouble understanding how the equations are created using piece-wise data.
1. How do you create the heaviside...
I need to use Laplace transforms to find the solution to this system of Heaviside functions but I'm not sure where to start because the two different x's in the system are confusing me.
Should I start by taking the laplace transforms of both sides where the laplace of H(t-1) = e-s/s
These...