Average of function (using dirac delta function)

In summary, to compute the average value of the function f(x) over the interval x ∈ [0, 8], we can use the equation ∫ dx δ(x-y) f(x) = f(y), where δ(x) is the Dirac δ-function and exp(iπ) = −1. By plugging in the given values, we get the average of f(x) = 1/8 ∫ from 0 to 8 δ(x-1) dx 16x2sin(πx/2)*(-1)/(1+x)(2-x). Simplifying this, we get an average of -1. This means that the average value of the function f(x) is -1 over the given
  • #1
j3dwards
32
0

Homework Statement


Compute the average value of the function:

f(x) = δ(x-1)*16x2sin(πx/2)*eiπx/(1+x)(2-x)

over the interval x ∈ [0, 8]. Note that δ(x) is the Dirac δ-function, and exp(iπ) = −1.

Homework Equations


∫ dx δ(x-y) f(x) = f(y)

The Attempt at a Solution


Average of f(x) = 1/8 ∫from 0 to 8 δ(x-1) dx 16x2sin(πx/2)*(-1)/(1+x)(2-x)
Average of f(x) = -1

Is this correct? I'm unsure of whether you can just use δ(x-a) = δ(x-1) and let a=1 and not let a=-1? I don't get how to use this bit of the function as I seem to have just ignored the negative sign.

Many thanks.
 
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  • #2
From the problem statement, your a is = to 1 not -1.
 

1. What is the Dirac delta function?

The Dirac delta function, denoted as δ(x), is a mathematical function that is zero for all values of x except at x = 0, where it is infinite. It is often used to model impulsive phenomena in physics and engineering.

2. How is the Dirac delta function used to calculate the average of a function?

The Dirac delta function can be used as a weighting function in an integral to calculate the average of a function. The integral of a function f(x) multiplied by the Dirac delta function δ(x) is equal to the average of the function f(x) over the domain of integration.

3. Can the Dirac delta function be used for continuous functions?

Yes, the Dirac delta function can be used for continuous functions. It is often used in conjunction with other functions in integrals to model impulsive phenomena in continuous systems.

4. Is the Dirac delta function the same as the Kronecker delta function?

No, the Dirac delta function and the Kronecker delta function are not the same. The Kronecker delta function, denoted as δij, is a discrete function that is equal to 1 when the indices i and j are equal, and 0 otherwise, whereas the Dirac delta function is a continuous function.

5. Can the Dirac delta function be used to find the average of a function over a specific interval?

Yes, the Dirac delta function can be used to find the average of a function over a specific interval. By multiplying the function f(x) by the Dirac delta function δ(x-a), where a is the starting point of the interval, and integrating over the entire domain, the average of the function over the interval can be calculated.

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