Integrating with a dirac delta function

In summary, to integrate \int_0^x \delta(x-y)f(y)dy, you can use the property of the Dirac Delta function that states \int_{a-ε}^{a+ε} f(y)\delta(y-a) \ dy = f(a), which results in the solution of f(x)(2H(x)-1).
  • #1
fred_91
39
0

Homework Statement



I have to integrate:

[itex]\int_0^x \delta(x-y)f(y)dy[/itex]

Homework Equations





The Attempt at a Solution



I know that the dirac delta function is zero everywhere except at 0 it is equal to infinity:
[itex]\delta(0)=\infty[/itex]
I have to express the integral in terms of function f only and i am unsure how to do this.
Do I have to use integration by parts?

Thank you very much in advance
 
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  • #2
fred_91 said:

Homework Statement



I have to integrate:

[itex]\int_0^x \delta(x-y)f(y)dy[/itex]

Homework Equations





The Attempt at a Solution



I know that the dirac delta function is zero everywhere except at 0 it is equal to infinity:
[itex]\delta(0)=\infty[/itex]
I have to express the integral in terms of function f only and i am unsure how to do this.
Do I have to use integration by parts?

Thank you very much in advance
Dirac's Delta is...weird. You have to integrate [itex]\int_0^x \delta(x-y)f(y)dy[/itex]?

Integration by parts sounds...okay? However, one of the properties of the Dirac Delta "function" is that [itex]\displaystyle \forall ε > 0, \int_{a-ε}^{a+ε} f(y)\delta(y-a) \ dy = f(a)[/itex]. Can you manipulate your integral into that form? :wink:

By my calculations, you should end up with [itex]f(x)(2H(x)-1)[/itex], where H is the antiderivative of δ.
 

1. What is a dirac delta function?

A dirac delta function, also known as the impulse function, is a mathematical function that is used to represent an infinitely high and narrow spike at a specific point. It is often used in physics and engineering to model point sources of energy or mass.

2. How is a dirac delta function integrated?

The dirac delta function is integrated using the sifting property, which states that the integral of the dirac delta function over a specific interval is equal to 1 if the interval contains the point of integration, and 0 if it does not. This means that the dirac delta function is only non-zero at the point of integration.

3. What is the purpose of integrating with a dirac delta function?

Integrating with a dirac delta function allows us to solve problems involving point sources or impulses. It also allows us to simplify complex integrals by replacing functions with delta functions, making the calculations easier to solve.

4. Can a dirac delta function be integrated over a finite interval?

No, the dirac delta function is only defined at a single point and has no value over a finite interval. It can only be integrated over the entire real line or over a specific interval that contains the point of integration.

5. How is the dirac delta function used in practical applications?

The dirac delta function is used in many practical applications, such as signal processing, control systems, and quantum mechanics. It is also used in solving differential equations and in the analysis of impulse responses in systems.

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