Sifting Property (Dirac Delta), please check these

The correct solution is e^(sqrt(2)/2). The rest of the solutions are correct.In summary, the sifting property of the Dirac delta function is used to evaluate various integrals, including those involving trigonometric functions and the exponential function. The correct solutions for the given integrals are 1, e^2, e^(sqrt(2)/2), and cos^(2)(y).
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Homework Statement


use the sifting property of the dirac delta function to evaluate the following integrals.

a) integral from -inf to inf sin(t) delta(t-pi/2)dt

b) integral from 0 to 2 e^(2t) delta(t-1)dt

c) integral from 0 to pi e^tan(theta) delta(theta- 3pi/4)d(theta)

d) integral from -inf to inf cos^(2)(theta) delta(theta+y)d(theta)

Homework Equations



integral from -inf to inf f(x) delta(x-x0)dx=f(x0)

The Attempt at a Solution



solutions:

a) 1
b) e^2
c) e^tan(theta)
d) cos^(2)(y)

are these solutions correct?
 
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  • #2
Your answer for c is wrong.
 

What is sifting property (Dirac Delta)?

The sifting property, also known as the Dirac delta function, is a mathematical concept used in engineering and physics to represent a point mass or impulse at a specific location. It is defined as an infinitely tall and narrow function with an area of one under the curve.

How is the sifting property (Dirac Delta) used?

The sifting property is used in many fields of science and engineering, particularly in signal processing and quantum mechanics. It is used to describe point sources and impulses in systems and to analyze and manipulate signals and waveforms.

What is the mathematical representation of the sifting property (Dirac Delta)?

The Dirac delta function is represented mathematically as δ(x-a), where a is the location of the point source or impulse. It is also defined as the limit of a sequence of functions that approach the delta function as their width approaches zero.

What are the properties of the sifting property (Dirac Delta)?

The sifting property has several key properties, including the sifting property itself (δ(x-a) = 1 when x=a and 0 otherwise), linearity, scaling, and time shifting. It also has a convolution property, which allows for the manipulation of signals through convolution with the delta function.

What are the limitations of the sifting property (Dirac Delta)?

While the sifting property is a useful mathematical concept, it has some limitations. It is not a function in the traditional sense, as it is not defined at a specific point. It also cannot be integrated in the traditional sense, and its use can lead to mathematical errors if not applied correctly.

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