- #1
SeM
Hi, if I have an interval on the x-axis, defined by the parameter L, can this, interval be transformed to a Dirac delta function instead, on the x-axis?
Thanks!
Thanks!
BvU said:Wouldn't it simply become a point ? Like in: the interval ##[x,x+\epsilon]## becomes the point ##x## for ##\epsilon \downarrow 0##
For an interval (a,b) where a<b, that would make L negative. If 'L' stands for length, do you mean L = b-a?SeM said:Thank you! This is indeed a great suggestion. However, L is included in a function I have, f(x), and currently it is a parameter. It would be useful to use your form somehow, as you write:
\begin{equation}
\int_{-\infty}^x \delta(t-a)dt - \int_{-\infty}^x \delta(t-b)dt
\end{equation}
Is L actually a-b here,
t is just a dummy variable for the integration, so it does not appear as an input or parameter of the indicator function.and t is the variable of my function, f(x)?
I'm confused about what you are trying to do here.If e^(ix)+L is my function, would the equivalent be :
\begin{equation}
\int_{-\infty}^x \delta(e^{it}-a)dt - \int_{-\infty}^x \delta(e^{it}-b)dt
\end{equation} ?
'It just says'? Where are you seeing this reference to the delta function? You need to back up and explain more about what your original problem is and why you want to use the delta function.SeM said:It just says \delta and it gives therefore no idea to me on how it looks like a function.
Who said that it is a 'dummy' function? I never said that.Nevertheless, you say the dirac f is a dummy function.
Why would you want to replace L with a delta function? Is L a constant?The thing I am not sure about is that because L is part of the existing function, f(x), say:
\begin{equation}
f(x) = e^{-ipk}/L
\end{equation}
how would the Dirac variant of this look like, where L is represented by the integral?
L is a parameter, the width of the space the particle is in.FactChecker said:Why would you want to replace L with a delta function? Is L a constant?
The Dirac delta function, also known as the unit impulse function, is a mathematical function that is defined as zero everywhere except at a single point, where its value is infinite. It is often used in engineering and physics to represent a concentrated point of mass, charge, or energy.
The Dirac delta function is typically represented by the symbol δ or δ(x). It is a continuous function that has a value of 0 for all values of x except at x = 0, where it has a value of infinity.
The Dirac delta function is used to model idealized point sources in physics and engineering problems. It allows for the simplification of mathematical equations by representing a point source as a function rather than a distribution of values.
The Dirac delta function is commonly used in calculus to represent discontinuous functions, such as step functions. It is also used to define the derivative of a step function, which is called the Heaviside step function.
Yes, the Dirac delta function can be integrated, but it requires the use of a special type of integration called the Dirac integral. The Dirac integral is defined as the limit of a sequence of approximating functions, and it allows for the integration of the Dirac delta function over a specific interval.