Delta Definition and 1000 Threads

Hello, Dirac Delta Function is defined as the function that its amplitude is zero everywhere except at zero where its amplitude is infinitely large such that the area under the curve is unity. Sometimes it is used to describe a function consists of a sequence of samples such as...

Using the defining property of the dirac delta function, \int{dx f(x) \delta(x-c)} show that \delta(ax)=\frac{1}{|a|}\delta(x) I think all I need to do is make the right u substitutions and it will come out right, but I can't think of how to make the substitutions...after a long time working...

Dirac delta function as the limit of a sequence Hi.. If I have a sequence which in some limit tends to infinity for x=0 and goes to zero for x\neq0, then can I call the limit as a dirac delta function? If not, what are the additional constraints to be satisfied?

Homework Statement Let V=x^3 Find dV and \Delta V. Show that for very small values of x , the difference \Delta V - dV is very small in the sense that there exists \varepsilon such that \Delta V - dV = \varepsilon \Delta x, where \varepsilon \to 0 as \Delta x \to 0.Homework Equations...

Claim: \nabla \cdot \frac{\hat{e}_r}{r^2}=4\pi\delta^3(\vec{x}) Anyone know of a proof of this? (or a reference which covers it?) We need to show that \frac{1}{4\pi}\int_0^R{(\nabla \cdot \frac{\hat{e}_r}{r^2})f(r)dr=f(0). The claimed identity can be seen in the solution for...

Homework Statement Evaluate the following integrals: \int^{+\infty}_{-\infty}\delta[f(x)]dx and \int^{+\infty}_{-\infty}\delta[f(x)]g(x)dx Homework Equations \int^{+\infty}_{-\infty}\delta(x)dx=1 \int^{+\infty}_{-\infty}\delta(x)f(x)dx=f(0) \int^{+\infty}_{-\infty}\delta(x-a)f(x)dx=f(a)The...

I know I haven't entered the formulae with the proper syntax, but I'm extremely exhausted at the time of posting, so please just read it and give advice, forgiving me this once for not using proper form (it's basically in latex code format). Homework Statement Show...

i know how to do basic proofs, but some proofs on the actual limit theorems confuse me. my textbook's choices for delta are very obscure and i have no idea how they even came up with them. for the proof of the limit theorem where the limit of a product of 2 functions is equal to the product...

I'm trying to understand the multiple limit processes involved with the Dirac delta function. Does it matter which process you do first the integral or the delta parameter that approaches zero? The closest theorem I found that addresses the order of taking limits is the Dominate Convergence...

We all know that \frac{1}{2\pi}\int{e^{ik(x-x')}dk=\delta(x-x'). i am working a problem which appears to depend on the statement \int e^{z^*(z-w)}dz^*\propto\delta(z-w) Does anyone know if this is valid? \delta(z-w) is defined in the usual way so that...

Claim: if \psi is a variable grassmann number, then \delta(\psi)=\psi is a Dirac delta function for integrals over \psi. I'm not seeing this. A general function of a grassmann number can be written f(\psi)=a+b\psi because anti-commutativity requires \psi^2=0. According to the wikipedia...

\begin{array}{l} \delta _{jk} A_k \\ \\ \delta _{jk} A_k = \left( {\delta _{1,1} + \delta _{1,2} + \delta _{1,3} + \delta _{2,1} + \delta _{2,2} + \delta _{2,3} + \delta _{3,1} + \delta _{3,2} + \delta _{3,3} } \right)A_k \\ \,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {1 + 0 + 0 +...

Hi all, I'm working through Chandrasekhar's http://prola.aps.org/abstract/RMP/v15/i1/p1_1" and can not understand the steps to progress through Eq. (66) in Chapter 1. The integral is: \prod^{N}_{j=1} \frac{1}{l^{3}_{j}|\rho|}\int^{\infty}_{0} sin(|\rho|r_{j})r_{j}\delta...

Is delta enthalpy zero for... Is delta H, for an isothermal process in which a gas expands? delta H=delta U+work done,correct? delta U is 0,but work done = nCln(V2/V1). So,delta H is not equal to 0.But my book says so.

Homework Statement http://img42.imageshack.us/img42/3537/powersyst.jpg Homework Equations P = I^2 R The Attempt at a Solution Here's what I have so far: I1 = 49.68A<-68.6 degrees (top loop) I2 = 49.68A<-128.6 degrees (bottom loop) I3 = 28.68<-98.6 degrees (triangle loop)...

I'm told that a product of distributions is undefined. See, http://en.wikipedia.org/wiki/Distribution_(mathematics)#Problem_of_multiplication where the Dirac delta function is considered a distribution. Now the Dirac delta function is defined such that, \[ \int_{ - \infty }^{ +...

Folks, I am looking for a blower to produce a flow of ~150cfm (~4m^3/min) while pumping H2 pressurized at ~40bar. The blower needs to provide a pressure increase of ~5bar. I am wondering if something like a supercharger blower might work.

I am confusing about the delta G(free-energy change). Could anyone explain me more about the sign of delta G. wat the exergonic and endergonic process mean? also, wat is the relation between the delta G and delta S(entropy)? I know the formula between them, but I don't quite understand Thank you,

why in the problem of dirac delta potential, the integral \int^{\epsilon}_{-\epsilon}\phi''(x)dx is equal to \phi'(\epsilon)-\phi'(-\epsilon)? but \int^{\epsilon}_{-\epsilon}\phi(x)dx is equal to 0 if, for example\phi(x)=e^x then \phi(x)''=\phi(x) but, the firts integral is...

Hi guys Can anybody help me? What is the difference between a delta \delta W and a differential dW? (W a scalar function, for example.) In other words, when shold be used a delta and when a differential? Thanks.

the integral \int_{-\infty}^{\infty} \! f(t)*\delta (t-t_0) * \delta (w-w(t)) \, dt is?? can be \delta (w-w(t)) * f(t_0) ?

Hello, I have just integrated over one variable, x and have now got a delta function \delta(m) where m = constant * (s-s') now I have to integrate over either s or s' but I am a bit confused since if I integrate over say s then the delta function depends on s. Hope I have explained clearly...

I do not know how to execute the problem with the 2x in the problem. Evaluate the integral: \int_{-4}^{4} (x^2+2x+1) \delta(2x) dx

1. Consider a steel guitar string of initial length L=1.00 meter and cross-sectional area A=0.500 square millimeters. The Young's modulus of the steel is Y=2.0 \times 10^{11} pascals. How far ( Delta L) would such a string stretch under a tension of 1500 Newtons? Use two significant figures in...

hey if lim (x-->0) f(x) = L where 0 < |x| < d1 implies |f(x) - L | < e how do i prove lim (x --> 0) f(ax) = L? i know 0 < |ax| < |a|d1 d2 = |a|d1 but the textbook says d2 = d1/|a| help you guyssssssssssssssssssssssssssssssss

Homework Statement Given f(x,y) = DeltaFunction(y - x*tan(theta)) a) Plot function. b) Take Fourier transform. c) Plot resulting transform. Homework Equations Delta function condition non-zero condition DeltaFunction(0) = Infinity Sifting property of delta functions The...

Given: limit of (sin x)/x as x --> 0 and that ε = .01 Problem: Find the greatest c such that δ between zero and c is good. Give an approximation to three decimal places. Equations: 0 < |x - a| < δ 0 < |f(x) - L| < εAttempt: 0 < |x - 0| < δ 0 < | sin(x)/x - 1| < ε 0 < | sin(x)/x - 1| < .01 0...

First of all, let me copy the standard solution from Griffiths, section 2.5, just for the sake of clarity. PotentialV(x) = - \alpha \delta (x) The bound state eigenfunction: \psi (x) = \left\{ \begin{array}{l} B{e^{\kappa x}}{\rm{ (}}x \le 0{\rm{)}} \\ B{e^{ - \kappa x}}{\rm{...

Given the limit of \frac{x^2+2x}{x^2-3x} as x approaches 0 equals \frac{-2}{3} and that ε = .01, find the greatest c such that every δ between zero and c is good. Give an exact answer. 0 < |x-0| < δ 0 < |\frac{x^2+2x}{|x^2-3x} + \frac{2}{3|}| < ε |\frac{x(x+2)}{|x(x-3)} +...

I had just reviewed back the properties of Delta Dirac Function, however I'm having a little confusing about the first property as stated : \delta\left(x-a)\right = 0 if x \neq a, \delta\left(x-a)\right = \infty if x = a;Here is my problem : when integrate over the entire region (ranging from...

Homework Statement Let f: \Re \rightarrow \Re and g: \Re \rightarrow \Re be functions such that lim_{x \rightarrow 1} f(x)=\alpha and lim_{x \rightarrow 1} g(x)=\beta for some \alpha, \beta \in \Re with \alpha < \beta . Use the \epsilon-\delta definition of a limit to prove...

I am trying to see why exactly the momentum eignenstates for a free particle are orthogonal. Simply enough, one gets: \int_{-\infty}^{\infty} e^{i (k-k_0) x} dx = \delta(k-k0) I can see why, if k=k0, this integral goes to zero. But if they differ, I don't see why it goes to zero. You have...

I am trying to integrate a charge density over a volume in order to obtain a total charge, but there is a delta function involved and I am not entirely sure how to treat it. \rho = q* \delta (\textbf{r})- \frac {q\mu^{2} Exp(- \mu r)} {4 \pi r} Q = \int \rho (\textbf{r})d^{3}r...

Homework Statement given a function defined by f(x,y) {= |xy|^a /(x^2+y^2-xy), if (x,y) cannot be (0,0) and = 0, if (x,y) = (0,0) Find all values of the real number a such that f is continuous everywhere e= epsilon d= delta In order to prove this, I know I need to do an...

Homework Statement What is the (volume) charge density of a ring of radius r_0 and uniform charge density \lambda? Homework Equations The Dirac Delta Function The Attempt at a Solution I've done a few line charge densities of straight wires along an axis (usually z, but on x as...

hello all, i am unaware of how to handle a delta function. from what i read online the integral will be 1 from one point to another since at zero the "function" is infinite. overall though i don't think i know the material well enough to trust my answer. and help on how to take the integral of...

Homework Statement I am really confused in my electrodynamics class. I have the following function. f(x) = \delta (x + \alpha ) + \delta(x -\alpha) How do i convert this into Fourier Tranform ? Those are dirac delta functions on either sides of the origin. Homework Equations...

hi, may someone help me to clarify my doubts... in my work, i encounter diracdelta square \delta(x-x_1)\delta(x-x_2) i am not sure what it means... it seems if i integrate it \int dx \;\delta(x-x_1)\delta(x-x_2) = \delta(x_1-x_2) is either zero of infinity. is this correct? thanks

For a field of 15 T, I calculated the magnitude of the splitting, which was 1.391E-22 J (this is delta E), i.e. delta E = |e| / 2m hbar B_z (m2 - m1) where m2 and m1 are the m_l levels. In order to determine the spacings for the visible lines on the absorption spectra, will that just be...

Homework Statement Suppose f(x) = x2 + x + 1, a = 1, and L = 3. Find a value d > 0 such that 0 < |x - a| < d implies |f(x) - L| < 1/100 Homework Equations The Attempt at a Solution Given 0<|x-1|<d implies 0<|x2 + x + 1 - 3|<1/100 0< x2 + x + -2 <1/100 0<(x+2)(x-1)<1/100 Assume 0<|x-1|<1...

Homework Statement Prove the following states directly using the formal e, d definition \lim_{x\rightarrow 8} \sqrt{x + 1} = 3 Homework Equations The Attempt at a Solution If 0 < |x-8| < d Then 0 < sqrt((x+1) - 3) < e Let e be given 3 < sqrt(x+1) < e + 3 9 < x + 1 <...

Homework Statement I wonder how to deal with the square root of Dirac Delta function, \sqrt{\delta(x)}. Actually, this comes from a problem which asking readers to calculate the wave function of a free particle and of a harmonic oscillator at time t, provided that the wave function at time...

Homework Statement On July 1, 2004, the Cassini spacecraft approached Saturn with hyperbolic excess velocity 5.5 km/s to swing by the planet at the closest approach distance rp = 80,680 km. Compute the impulsive ΔV required for a maneuver performed at the closest approach to Saturn to...

Whats the difference between average velocity and delta velocity? I understand the formulas (ie. Vave = delta displacement/delta time) but I don't understand why you would use one over the other and what the difference is. Thanks for the help.

Homework Statement Suppose A_n is, for all natural numbers n, some finite set of numbers in [0,1] and A_n intersect A_m={ } if m!=n Define f as follows: f(x) = 1/n if x is in A_n and 0 if x is not in A_n for all n. Prove that the limit as x goes to a of f(x) = 0 for all a in [0,1]...

I am supposed to prove that δ'(ax) = (1/a)*(1/|a|)*δ'(x) but I cannot figure out where the (1/a) term comes from. Using the scaling theorem I know that δ(ax) = (1/|a|)*δ(x), but how does this apply to the first derivative and does it explain where the (1/a) comes from?

The problem is to prove that δ'(ax) = (1/a)*(1/a)*δ'(x), where a is a constant. I tried applying the scaling theorem with the formal definition of δ'(x) but I can not get the second (1/a) term. Does anyone have some insight on this problem? Thank you...

Awesome vantage point.

Homework Statement Can anyone remember a decent argument/derivation for the following representation of the delta function. Homework Equations $ \nabla^2 \frac{1}{|r|} =\delta(r)$ (probally up to some multipicative constant $\frac{1}{2\pi}$ or something The Attempt at a...

How would you write an infinite line charge with constant charge per unit length \lambda as a volume charge density using Dirac delta functions? Perhaps in cylindrical coordinates? I'm confused because if you integrate this charge distribution over all space, you should get an infinite amount...