Solving Dirac Delta Cosx: Find Range of n and a_n, x_n

In summary, the function \delta(cosx) can be written as a sum of Dirac delta functions, with n ranging to infinity. The values for a_n are equal to the roots of cos(x)=0, and x_n is the sum of the delta functions divided by the absolute value of the derivative of cos(x).
  • #1
apw235
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Homework Statement


The function [tex]\delta(cosx)[/tex] can be written as a sum of Dirac delta functions:
[tex]\delta(cosx)=\sum_{n} a_{n}\delta(x-x_{n})[/tex]
Find the range for n and the values for [tex]a_{n}[/tex] and [tex]x_{n}[/tex]


The Attempt at a Solution


Well, taking the integral of [tex]\delta(cosx)[/tex], we only get spikes when x is an even multiple of [tex]\frac{\pi}{2}[/tex]. So shouldn't n run to infinity? Thats all i have so far, any help would be appreciated. thanks.

-Adrian
 
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  • #2
You get 'spikes' where cos(x)=0. I wouldn't describe those as 'even multiples of pi/2'. In general if x_i are the roots of f(x)=0, then delta(f(x)) is the sum of delta(x_i)/|f'(x_i)|.
 

1. What is the Dirac Delta function?

The Dirac Delta function, denoted by δ(x), is a mathematical function that is commonly used in physics and engineering to represent a point mass or impulse at a specific location. It is defined as zero everywhere except at x = 0, where it is infinite, with an area under the curve of exactly 1.

2. How do you solve for the Dirac Delta function in cosx?

To solve for the Dirac Delta function in cosx, we use the formula ∑ ancos(nx) = δ(x) where an represents the Fourier coefficients. By plugging in the desired function, cosx, and solving for the Fourier coefficients, we can find the values of n and an that satisfy the equation.

3. What is the range of n in solving for Dirac Delta in cosx?

The range of n in solving for Dirac Delta in cosx depends on the desired accuracy and precision of the solution. In general, the larger the range of n, the more accurate the solution will be. However, for practical purposes, a range of n between -10 and 10 is often sufficient.

4. How do you find the values of an in solving for Dirac Delta in cosx?

The values of an can be found by using the formula an = (1/2π) ∫ cos(nx)δ(x)dx, where ∫ represents the integral and dx is the differential of x. This integral can be solved using various techniques, such as integration by parts or the Fourier transform.

5. What is the significance of solving for Dirac Delta in cosx?

Solving for Dirac Delta in cosx allows us to represent a complex function, such as cosx, as a series of simpler functions, making it easier to analyze and manipulate in mathematical calculations. This technique is also commonly used in signal processing and other applications in physics and engineering.

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