Understanding Dirichlet Integral Proof

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In summary, the conversation discusses the proof given in a Wikipedia page about the Dirichlet integral. One person is unsure about a step involving the sine being the imaginary part of the exponential and expanding it to the whole integral. Another person explains that the step is valid and can be solved using integration by parts, but also notes that both alpha and omega must be real for it to work. The conversation ends with one person thanking the others for their help.
  • #1
Amok
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I was trying to understand the proof given in this wiki page:

http://en.wikipedia.org/wiki/Dirichlet_integral

But I'm sure their proof is correct because I don't know if the step where he says the sine is the imaginary part of the the exponential and then expands it to the whole of the integral is correct.

[tex]\int_0^{\infty} e^{-\alpha \omega} Im( e^{i \beta \omega}) = Im( \int_0^{\infty} e^{-\alpha \omega} e^{i \beta \omega})[/tex]

I doesn't make much sense to me. That whole part of the demonstration is very unclear to me.

Anyone know this?
 
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  • #2
[tex]e^{-\alpha\omega}[/tex] is real there, and since [tex]c \Im \left{a+bi\right}=\Im\left{ac+cbi\right}=cb[/tex], the equality [tex]e^{-\alpha\omega}\sin{\beta\omega}=\Im\left{e^{-\alpha\omega}e^{i\beta\omega}\right}[/tex] is valid. If you are still uncertain of this, the integral can be solved by using twice integration by parts.
 
  • #3
losiu99 said:
[tex]e^{-\alpha\omega}[/tex] is real there, and since [tex]c \Im \left{a+bi\right}=\Im\left{ac+cbi\right}=cb[/tex], the equality [tex]e^{-\alpha\omega}\sin{\beta\omega}=\Im\left{e^{-\alpha\omega}e^{i\beta\omega}\right}[/tex] is valid. If you are still uncertain of this, the integral can be solved by using twice integration by parts.

So alpha and omega have to be real for this to work.
 
  • #4
Yes. Also, don't forget your differentials.
 
  • #5
Yeah, well they do not state that alpha is real in that page. Thanks guys.
 

1. What is a Dirichlet Integral?

A Dirichlet Integral is a type of integral that is used to evaluate the convergence of a series. It was first introduced by the mathematician Peter Gustav Lejeune Dirichlet in the 19th century.

2. How is a Dirichlet Integral calculated?

The Dirichlet Integral is calculated by taking the limit of a partial sum of a series as the number of terms in the series approaches infinity. It is represented mathematically as ∫₀ᵃ f(x)dx = lim(n→∞)∑(k=0 to n) f(ak)Δx, where a is the starting point, f(x) is the integrand, and Δx is the step size.

3. What is the significance of the Dirichlet Integral in mathematics?

The Dirichlet Integral is important in mathematics because it is used to test the convergence of a series, which is a fundamental concept in calculus and analysis. It is also used in various fields such as physics, engineering, and economics.

4. What is the proof of the convergence of Dirichlet Integral?

The proof of the convergence of Dirichlet Integral involves using the Cauchy Criterion, which states that a series converges if and only if the limit of the partial sums approaches a finite number. The proof also involves using the Dirichlet Test, which provides sufficient conditions for the convergence of a series.

5. How can understanding Dirichlet Integral be applied in real-world scenarios?

Understanding Dirichlet Integral can be applied in real-world scenarios such as analyzing the stability of economic systems, determining the convergence of electrical circuits, and calculating the rate of heat transfer in thermodynamics. It can also be used in signal processing, control theory, and image processing.

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