Difficult integral

  • #1
Mod note: Moved from the math technical sections.
I need to show that
[tex]\int_{-\infty}^\infty \frac{\sin^2 (pa/\hbar)}{p^2} \, dp = \frac{\pi a}{\hbar}.[/tex]
I haven't got a clue how to integrate this function! Any help would be much appreciated thanks.
 
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Answers and Replies

  • #2
I've found through the transformation [itex]u=pa/\hbar[/itex] that it is equivalent to showing
[tex] \int_{-\infty}^{\infty} \frac{\sin^2 u}{u^2}\,du = \pi,[/tex] if that helps anyone.
 
  • #3
SteamKing
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You've got your substitution and your integral.

All you have to do is figure out p and dp in terms of u and du.

The algebra is really simple.
 
  • #4
You've got your substitution and your integral.

All you have to do is figure out p and dp in terms of u and du.

The algebra is really simple.

Yeah I did that substitution myself to make it easier... now how do I show that the integral is equal to pi??
 
  • #5
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Using the cosine of double angle formula, and then integration by parts, this can be reduced to integrals of $$ \frac {\sin x} {x} $$
 
  • #6
65
0
Under a change of variable x = pa/h:

[tex]\frac{\hbar}{a}\int_{-\infty}^\infty \frac{sin^2 x}{x^2}dx[/tex]

As suggested you can apply integration by parts and then sine double angle rule to obtain

[tex]\frac{\hbar}{a}\int_{-\infty}^\infty \frac{sin(2x)}{x}dx[/tex]

The integral below has many different proofs, few of them are here
[tex]\int_{-\infty}^\infty \frac{sin(ωx)}{x}dx=\pi[/tex] for any ω > 0.
 
  • #7
64
1
OFF-TOPIC:

Can someone tell me what is the difference between a question asked here and a question asked on the homework forums?

This is an honest question. For example, shouldn't this question ("Difficult integral") be in the homework section?

Sorry for the off topic perishingtardi
 
  • #8
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Crake,
You are correct. As the sticky says at the top of this forum section, "This forum is not for homework or any textbook-style questions."

I am moving this thread to the Homework section.
 

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