Dirac delta function Definition and 199 Threads

Hi - firstly should I be concerned that the dirac function is NOT periodic? Either way the problem says expand $\delta(x-t)$ as a Fourier series... I tried $\delta(x-t) = 1, x=t; \delta(x-t) =0, x \ne t , -\pi \le t \le \pi$ ... ('1' still delivers the value of a multiplied function at t)...

Ok so for equations of spherical wave in fluid the point source is modeled as a body force term which is given by time dependent 3 dimensional dirac delta function f=f(t)δ(x-y) x and y are vectors. so we reach an equation with ∫f(t)δ(x-y)dV(x) over the volume V. In the textbook it then says that...

1. Homework Statement Find the Fourier series of ##f(x) = \delta (x) - \delta (x - \frac{1}{2})## , ## - \frac{1}{4} < x < \frac{3}{4}## periodic outside. Homework Equations [/B] ##\int dx \delta (x) f(x) = f(0)## ##\int dx \delta (x - x_0) f(x) = f(x_0)##The Attempt at a Solution...

I consider the Dirac delta. In physics the delta squared has an infinite norm : $$\int\delta (x)^2=\infty $$ But if i look at delta being a functional i could write : $$\delta [f]=f (0) $$ hence $$\delta^2 [f]=\delta [\delta [f]]=\delta [\underbrace {f (0)}_{constant function}]=f (0)$$ Thus...

Homework Statement a) and b) are no problem. I need help to solve c) and d) Homework Equations c) Delta dirac function Gauss' law d) Gauss' law ## \int_V {\rho \, d\tau} = Q_{enclosed} ## The Attempt at a Solution By taking laplace on the potential I get: ## \rho(\mathbf{r}) =...

I have recently digged up a post in the forum about a confusion arise from definition of Dirac Delta function and I am actually really bothered by it (link to the thread). When people talk about sampling some function f(x) with Dirac Comb, or impulse train, they would be talking about the...

Hello community, this is my first post and i start with a question about the famous dirac delta function. I have some question of the use and application of the dirac delta function. My first question is: Using Dirac delta functions in the appropriate coordinates, express the following charge...

Homework Statement Hi all, I'm currently reviewing for a final and would like some help understanding a certain part of this particular problem: Determine the retarded Green's Function for the D'Alembertian operator ##D = \partial_s^2 - \Delta##, where ##\Delta \equiv \nabla \cdot \nabla## ...

I've been thinking about the properties of the Dirac delta function recently, and having been trying to prove them. I'm not a pure mathematician but come from a physics background, so the following aren't rigorous to the extent of a full proof, but are they correct enough? First I aim to...

For a research project, I have to take multiple derivatives of a Yukawa potential, e.g. ## \partial_i \partial_j ( \frac{e^{-m r}}{r} ) ## or another example is ## \partial_i \partial_j \partial_k \partial_\ell ( e^{-mr} ) ## I know that, at least in the first example above, there will be a...

Homework Statement I am having trouble understanding this: I have a Dirac Delta function $$ \delta (t_1-t_2) $$ but I want to prove that in the frequency domain (Fourier Space), it is: $$\delta(\omega_1+\omega_2) $$ Would anyone have any ideas how to go about solving this problem? I know...

hi deoes anyone know any online resource for proofs of Dirac delta function identities and confirming which representations are indeed DD functions Thanks a lot.

Homework Statement Problem: a) Find the Fourier transform of the Dirac delta function: δ(x) b) Transform back to real space, and plot the result, using a varying number of Fourier components (waves). c) test by integration, that the delta function represented by a Fourier integral integrates...

For proving this equation: \delta (g(x)) = \sum _{ a,\\ g(a)=0,\\ { g }^{ ' }(a)\neq 0 }^{ }{ \frac { \delta (x-a) }{ \left| { g }^{ ' }(a) \right| } } We suppose that g(x)\approx g(a) + (x-a)g^{'}(a) Why for Taylor Expansion we just keep two first case and neglect others...

So part of the idea presented in my book is that: div(r/r3)=0 everywhere, but looking at this vector field it should not be expected. We would expect some divergence at the origin and zero divergence everywhere else. However I don't understand why we would expect it to be zero everywhere but...

How can I find the derivative of a step function?

How to calculate ##\int^{\infty}_{-\infty}\frac{\delta(x-x')}{x-x'}dx'## What is a value of this integral? In some youtube video I find that it is equall to zero. Odd function in symmetric boundaries.

I have a Gaussian distribution about t, say, N(t; μ, σ), and a a Dirac Delta Function δ(t). Then how can I compute: N(t; μ, σ) * δ(t > 0) Any clues? Or recommender some materials for me to read? Thanks!

Sorry if the question seems naive but if we have the Dirac delta function delta(x-y) is it the same as delta(y-x)?? Or there are opposite in sign? And why ? Thank you for your time

If you ask me define Dirac delta function, i can easily define it and prove its properties using laplacian or complex analysis method. But still i don't understand what is the use of DIRAC DELTA FUNCTION in quantum mechanics. As i have done some reading Quantum mechanics from Introduction to...

Homework Statement Find the inner product of f(x) = σ(x-x0) and g(x) = cos(x) Homework Equations ∫f(x)*g(x)dx Limits of integration are -∞ to ∞ The Attempt at a Solution First of all, what is the complex conjugate of σ(x-x0)? Is it just σ(x-x0)? And I'm not sure how to...

Homework Statement OK so I'm doing a course on Signals and Systems and I'm taking inverse z transforms using residue integration. One particular formula in complex integration made me think a bit. \oint{\frac{f(z)}{z-z_0} dz} = 2\pi jf(z_0) This looks eerily similar to the definition...

Homework Statement The Potential V(r) is given: A*e^(-lambda*r)/r, A and lambda are constants From this potential, I have to calculate: E(r), Rho(r) -- charge density, and Q -- total charge. Homework Equations The Attempt at a Solution I know that E(r) is simply minus...

Homework Statement Prove this theorem regarding a property of the Dirac Delta Function: $$\int_{-\infty}^{\infty}f(x)\delta'(x-a)dx=-f'(a)$$ (by using integration by parts) Homework Equations We know that δ(x) can be defined as...

Does the Dirac delta have a residue? It seems like it might, but I don't know how to attack it, since I really know very little about distributions. For example, the Dirac delta does not have a Laurent-expansion, so how would you define its residue?

What does the Dirac Delta Function do? ##\delta^3(\vec{r})## How do you evaluate it? What are its values from -inf to +inf?

It's been quite some time now since I decided to stop self-studying physics and to pay more attention to the math behind. I'm working towards gaining an understanding of 100% rigorous mathematics for now. One thing that has always bothered me is the Dirac delta function. What I want to know...

Homework Statement Good day. May I know, for Dirac Delta Function, Is δ(x+y)=δ(x-y)? The Attempt at a Solution Since δ(x)=δ(-x), I would say δ(x+y)=δ(x-y). Am I correct?

From what I can tell, it seems that 1/x + δ(x) = 1/x because if we think of both 1/x and the dirac delta function as the following peicewise functions: 1/x = 1/x for x < 0 1/x = undefined for x = 0 1/x = 1/x for x > 0 δ(x) = 0 for x < 0 δ(x) = undefined for x = 0 δ(x) = 0 for x > 0...

Homework Statement Consider the double Dirac delta function V(x) = -α(δ(x+a) + δ(x-a)). Using this potential, find the (normalized) wave functions, sketch them, and determine the # of bound states. Homework Equations Time-Independent Schrodinger's Equation: Eψ = (-h^2)/2m (∂^2/∂x^2)ψ +...

In page 555, Appendix B of Intro to electrodynamics by D Griffiths: \nabla\cdot \vec F=-\nabla^2U=-\frac{1}{4\pi}\int D\nabla^2\left(\frac{1}{\vec{\vartheta}}\right)d\tau'=\int D(\vec r')\delta^3(\vec r-\vec r')d\tau'=D(\vec r) where ##\;\vec{\vartheta}=\vec r-\vec r'##. Is it supposed to be...

I want to proof (1)##\delta(x)=\delta(-x)## and (2) ## \delta(kx)=\frac{1}{|k|}\delta(x)## (1) let ##u=-x\Rightarrow\;du=-dx## \int_{-\infty}^{\infty}f(x)\delta(x)dx=(0) but \int_{-\infty}^{\infty}f(x)\delta(-x)dx=-\int_{-\infty}^{\infty}f(-u)\delta(u)du=-f(0) I cannot proof (1) is equal as I...

My question is in Griffiths Introduction to Electrodynamics 3rd edition p48. It said Two expressions involving delta function ( say ##D_1(x)\; and \;D_2(x)##) are considered equal if: \int_{-\infty}^{\infty}f(x)D_1(x)dx=\int_{-\infty}^{\infty}f(x)D_2(x)dx\;6 for all( ordinary) functions f(x)...

I know this probably belongs in one of the math sections, but I did not quite know where to put it, so I put it in here since I am studying Electrodynamics from Griffiths, and in the first chapter he talks about Dirac Delta function. From what I've gathered, Dirac Delta function is 0 for...

Hello, Is this correct: \int [f_j(x)\delta (x-x_i) f_k(x)\delta (x-x_i)]dx = f_j(x_i)f_k(x_i) If it is not, what must the left hand side look like in order to obtain the right handside, where the right hand side multiplies two constants? Thanks!

Hi there, I'm trying to comprehend Dirac Delta functions. Here's something to help me understand them; let's say I want to formulate Newton's second law F=MA (for point masses) in DDF form. Is this correct: F_i = \int [m_i\delta (x-x_i) a_i\delta (x-x_i)]dx Or is it this: F_i = [\int...

Homework Statement I am trying to integrate the function \int _{-\infty }^{\infty }(t-1)\delta\left[\frac{2}{3}t-\frac{3}{2}\right]dt Homework Equations The Attempt at a Solution I think the answer should be \frac{5}{4} because \frac{2}{3}t-\frac{3}{2}=0 when t=9/4. then (9/4-1) = 5/4...

Hi all, I'm familiar with the fact that the dirac delta function (when defined within an integral is even) Meaning delta(x)= delta(-x) on the interval -a to b when integral signs are present I want to prove this this relationship but I don't know how to do it other than with a limit...

Hi All, I have a problem in understanding the concept of dirac delta function. Let say I have a function, q(r,z,t) and its defined as q(r,z,t)= δ(t)Q(r,z), where δ(t) is dirac delta function and Q(r,z) is just the spatial distribution. My question are: 1. How can I find the time derivative...

I have been wondering exactly how one would express the Dirac delta in arbitrary spaces with curvature. And that leads me to ask if the Dirac delta function has a coordinate independent expression. Is there an intrinsic definition of a Dirac delta function free of coordinates and metrics? Or as...

Homework Statement I have to integrate: \int_0^x \delta(x-y)f(y)dy Homework Equations The Attempt at a Solution I know that the dirac delta function is zero everywhere except at 0 it is equal to infinity: \delta(0)=\infty I have to express the integral in terms of function...

Homework Statement L[t^{2} - t^{2}δ(t-1)] Homework Equations L[ t^{n}f(t)] = (-1^{n}) \frac{d^{n}}{ds^{n}} L[f(t)] L[δ-t] = e^-ts The Attempt at a Solution My teacher wrote \frac{2}{s^{3}} -e^{s} as the answer. I got \frac{2}{s^{3}} + \frac{e^-s}{s} + 2 \frac{e^-s}{s^2} + \frac{2e^-s}{s^3}

Homework Statement The question was way too long so i took a snap shot of it http://sphotos-h.ak.fbcdn.net/hphotos-ak-snc7/397320_358155177605479_1440801198_n.jpg Homework Equations The equations are all included in the snapshotThe Attempt at a Solution So for question A I've done what the...

Homework Statement This is an issue I'm having with understanding a section of maths rather than a coursework question. I have a stage of the density function on the full phase space ρ(p,x); ρ(p,x) = \frac {1}{\Omega(E)} \delta (\epsilon(p,x) - E) where \epsilon(p,x) is the...

Prove that x \frac{d}{dx} [\delta (x)] = -\delta (x) this is problem 1.45 out of griffiths book by the way. Homework Equations I attempted to use integration by parts as suggest by griffiths using f = x , g' = \frac{d}{dx} This yields x [\delta (x)] - \int \delta (x)dx next I tried...

The Dirac delta "function" is often given as : δ(x) = ∞ | x = 0 δ(x) = 0 | x \neq 0 and ∫δ(x)f(x)dx = f(0). What about δ(cx)? By u=cx substitution into above integral is, ∫δ(cx)f(x)dx = ∫δ(u)f(u/c)du = 1/c f(0). But intuitively, the graph of δ(cx) is the same as the graph of...

Is it possible to take the variation of the Dirac delta function, by that I mean take the functional derivative of the Dirac delta function?

Hello I'm trying to figure out how to evaluate(in the distribution sense) \delta'(g(x)). Where \delta(x) is the dirac delta function. Please notice that what I want to evaluate is not \frac{d}{dx}(\delta(g(x))) but the derivative of the delta function calculated in g(x). If anyone could post...

hi! i have a question regarding the delta function. if i have a delta distribution with an argument that is a function of multiple arguments, somthimg like: ∫δ(E-p^{2}_{i}/2m)dp^{N}, ranging over +-∞ now, the argument of the delta function vanishes on a sphere. i can evaluate the...

Hi there, I am trying to integrate this: http://imm.io/oqKi I should get the second line from the integral, but I can't show it. This should somehow relate to the Heaviside step function, or I am completely wrong. Any ideas? Sorry about the url, I fixed it.