Strange representation of Heaviside and Delta function

In summary, the conversation discusses the representation of the Dirac delta function and a similar formula for the Heaviside function using contour integration. The conversation questions the validity and conditions of this approach.
  • #1
mhill
189
1
in the .pdf article http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6027/1/jfs080104.pdf

i have found the strange representation

[tex] \delta (x) = -\frac{1}{2i \pi} [z^{-1}]_{z=x} [/tex]

and a similar formula for Heaviside function replacing 1/z by log(-z) , what is the meaning ? of this formula
 
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  • #2
Probably the idea is, that we make it a contour integration. Then the only pole is at z = 0 and the result of an integration would be e.g.
[tex]\int \delta(x) f(x) \, dx = \int \frac{-1}{2i\pi} \frac{f(z)}{z} \, dz = (2i\pi) \operatorname{Res}_{z = 0} \frac{f(z)}{2 \pi i} = f(0)[/tex]
if f(x) does not have any poles in the upper half plane.
but I actually doubt how valid this is (even if it works, one would need requirements on f(x) for [itex]x \to \pm i\infty[/itex] to close the countour; and I wonder what happens if f(x) itself has poles).

So I think that is the idea, but I wonder if it works and under what conditions.
 

1. What is the Heaviside function and how is it represented?

The Heaviside function, also known as the unit step function, is a mathematical function that is defined as 0 for negative inputs and 1 for positive inputs. It is commonly represented as H(x) or u(x) in mathematical equations.

2. How does the Heaviside function relate to the Delta function?

The Heaviside function is closely related to the Delta function, also known as the Dirac delta function. The Delta function is defined as an infinitely narrow, infinitely tall spike at the origin, with an area of 1. The Heaviside function can be thought of as the integral of the Delta function, and is commonly used to "turn on" or "turn off" other functions in mathematical equations.

3. What are some real-world applications of the Heaviside and Delta functions?

The Heaviside and Delta functions have many applications in physics and engineering. They are commonly used in circuit analysis to model the behavior of electrical systems, in signal processing to represent impulses and discontinuities, and in control systems to describe the response of a system to an input signal.

4. Can the Heaviside and Delta functions be graphed?

While the Delta function cannot be graphed in the traditional sense due to its infinitely narrow width, the Heaviside function can be graphed as a step function. The graph of the Heaviside function has a horizontal line at y=0 for all negative x values, and a horizontal line at y=1 for all positive x values.

5. Are there any alternative representations of the Heaviside and Delta functions?

Yes, there are several alternative representations of the Heaviside and Delta functions, such as the sign function and the rectangular pulse function. These alternative representations may be more convenient in certain situations, but ultimately they all describe the same mathematical concept.

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