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?
I got the function f(x)=e−|x|f(x)=e−|x| I want to show that f′′(x)=f(x)−2δ(x)f″(x)=f(x)−2δ(x) where δ(x)δ(x) is the Dirac delta function. I know that I can solve it with a known theorem but can I prove it without using it? dirac-delta Sha...
Learn the definition of Dirac delta functions and browse a collection of 17 enlightening community discussions around the topic.
Definition : Properties of the delta function We define the delta function δ(x)δ(x) as an object with the following properties: δ(x)={∞0x=0otherwiseδ(x)={∞x=00otherwise δ(x)=ddxu(x)δ(x)=ddxu(x), where u(x)u(x) is the unit step function (Equation 4.8); ∫ϵ...
??)dA(0) Two questions arise: 1) Is the limiting process used above valid in the context of a Fourier-Stieltjes integral? 2) If the limit is valid, does the Dirac delta function operate on a Fourier-Stieltjes integral the same as it operates on a Reimann integral?
是由转置后取共轭,又因为proof:X†是由X转置后取共轭,又因为 |β⟩⟨α|=(β1β2)(α1∗α2∗)=(β1α1∗β1α2∗β2α1∗β2α2∗) |α⟩⟨β|=(α1α2)(β1∗β2∗)=(β1∗α1β2∗α1β1∗α2β2∗α2) 不难发现,两者是转置后共轭的存在,故得证...
Thus, we can express \(\psi _{\pm }^{\mu \nu }\) (and, finally, the current) via the scalar function \(\psi _{\mp }^{\mu \nu }u_{\mu \nu }\) (and its derivatives), and the results do not depend on the choice of the value of the square root in (62), as, for ...
Dirac delta function approach can also be useful for any kinds of broken-line functions. As an example, we represent a simple proof for the identity relating prime counting function and Li-function [11] ( ) ( ) ( ) ( ) 2 2 1
Prove that derivative of the theta function is the dirac delta function let θ(x-x') be the function such that θ = 1 when x-x' > 0 and θ = 0 when x-x' < 0. Show that d/dx θ(x-x') = δ(x - x'). it is easy to show that d/dx θ(x-x') is 0 everywhere excep...
. the function \(t \mapsto \text {im}\, ( f_{\pm , t} | d_{\omega ,\lambda } f_{\pm , t} )\) is continuous and converges to zero for \(t \rightarrow \infty \) , so for every \(\epsilon \in ]0, \delta ]\) there exists \(t \ge t_0\) such that \(\...