Showing the Representation of the Delta Function

In summary, the delta function is a mathematical concept used to represent a point mass, with a value of zero everywhere except at a single point. It is defined as having a value of infinity at x=0 and a value of zero everywhere else. Graphically, it is represented as a spike or impulse at x=0. The delta function has many real-life applications in physics, engineering, and signal processing, and is related to other mathematical concepts such as the unit step function and the Kronecker delta.
  • #1
LagrangeEuler
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Homework Statement


Show that
##\frac{1}{\pi}\lim_{\epsilon \to 0^+}\frac{\epsilon}{\epsilon^2+k^2}##
is representation of delta function.

Homework Equations


##\delta(x)=\frac{1}{2 \pi}\int^{\infty}_{-\infty}dke^{ikx}##

The Attempt at a Solution



##\int^{\infty}_{-\infty}\frac{\epsilon}{\epsilon^2+k^2}dk=\pi##
One can take ##F[e^{-\epsilon x}]## and then put ##\epsilon to go to zero +. Why ##0^+##. I'm confused?
 
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  • #2
Why ##0^+##. I'm confused?
What happens when you take the limit from the other side?
 

FAQ: Showing the Representation of the Delta Function

1. What is the purpose of showing the representation of the delta function?

The representation of the delta function is used to describe a mathematical concept known as a point mass, which is a function that has a value of zero everywhere except at a single point. This is useful in many areas of science, including physics, engineering, and signal processing.

2. What is the mathematical definition of the delta function?

The delta function, denoted as δ(x), is a special function that is defined as having a value of infinity at x=0 and a value of zero everywhere else. It is also known as the Dirac delta function, after the physicist Paul Dirac who first introduced it.

3. How is the delta function represented graphically?

The delta function is usually represented graphically as a spike or impulse at x=0, with a height of infinity. This represents the function's property of being zero everywhere except at a single point. In some cases, the spike may be replaced with a narrow rectangle with a height of infinity and width of zero.

4. What are some real-life applications of the delta function?

The delta function has many practical applications in science and engineering. It is commonly used to model point sources of energy or mass in physics, such as a point charge or a point mass. It is also used in signal processing to represent impulses or sudden changes in a signal.

5. How is the delta function related to other mathematical concepts?

The delta function has connections to other mathematical concepts, such as the unit step function, which is defined as the integral of the delta function. It is also closely related to the Kronecker delta, which is a discrete equivalent of the delta function. Additionally, the Fourier transform of the delta function is equal to a constant, making it useful in solving differential equations.

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