Difficult Integration

Printable View

November 8th 2006, 09:40 AM
topsquark

Difficult Integration

(sigh) I know I've seen how to do this before, but I can't find the notes! :mad:

The problem is this:
$\int_0^{\pi} d \theta \, sin \theta \, e^{ia \, sin \theta}$ where a is real.

It is simple enough to show that the integral is equal to a real number, so
$\int_0^{\pi} d \theta \, sin \theta \, e^{ia \, sin \theta} = \int_0^{\pi} d \theta \, sin \theta \, cos(a \, sin \theta)$
as an alternate formula.

Comparing two lines in the book I got this from the integral should be equal to
$2 \frac{sin \, a}{a}$
(For the case a = 0, the integral is simple and comes out to be equal to 2. Obviously I'm not worried about this case.)

Any takers? Thanks in advance.

-Dan
November 8th 2006, 11:22 AM
Quick

You should know that the answer is:
$\int_0^{\pi} d \theta \, sin \theta \, e^{ia \, sin \theta}=2$

You should also know never to listen to me when you get past algebra :D
November 8th 2006, 12:47 PM
topsquark

Quote:

Originally Posted by Quick

You should know that the answer is:
$\int_0^{\pi} d \theta \, sin \theta \, e^{ia \, sin \theta}=2$

You should also know never to listen to me when you get past algebra :D

No no no! When it's Calculus the answer is always 8.

-Dan
November 8th 2006, 06:37 PM
Quick

Quote:

Originally Posted by topsquark

No no no! When it's Calculus the answer is always 8.

-Dan

Oh, sorry you're right :D (I really need to review my notes)
November 8th 2006, 09:30 PM
CaptainBlack

1 Attachment(s)

Quote:

Originally Posted by topsquark

(sigh) I know I've seen how to do this before, but I can't find the notes! :mad:

The problem is this:
$\int_0^{\pi} d \theta \, sin \theta \, e^{ia \, sin \theta}$ where a is real.

It is simple enough to show that the integral is equal to a real number, so
$\int_0^{\pi} d \theta \, sin \theta \, e^{ia \, sin \theta} = \int_0^{\pi} d \theta \, sin \theta \, cos(a \, sin \theta)$
as an alternate formula.

Why do you think the integral is real?

See attachment.

RonL
November 9th 2006, 03:21 AM
topsquark

Quote:

Originally Posted by CaptainBlack

Why do you think the integral is real?

See attachment.

RonL

Heavens. I had transformed the integral by substituting $\theta' = \theta - \pi/2$ and took a look at $sin(acos(\theta'))$ and decided it was an odd function over $[-\pi/2, \pi/2]$ . It isn't, of course. :o

Now I've got a "real" problem (pun intended) because my book clearly indicates the real.

Thanks! Back to the books to find out what's going on.

-Dan
November 15th 2006, 04:35 PM
topsquark

Okay, I fixed my boo-boo. I wish I could say I used the wrong coordinate system or something face-saving, but I simply used the wrong trig function. :o

The correct integral is:
$\int_0^{\pi} d \theta \, sin \theta \, e^{ia \, cos \theta} = \int_0^{ \pi} d \theta \, sin \theta \, cos(a \, cos \theta) = 2 \frac{sin(a)}{a}$ where $a > 0$ .

Anyone know how to do this one? Thanks!

-Dan
November 15th 2006, 07:48 PM
CaptainBlack

Quote:

Originally Posted by topsquark

Okay, I fixed my boo-boo. I wish I could say I used the wrong coordinate system or something face-saving, but I simply used the wrong trig function. :o

The correct integral is:
$\int_0^{\pi} d \theta \, sin \theta \, e^{ia \, cos \theta} = \int_0^{ \pi} d \theta \, sin \theta \, cos(a \, cos \theta) = 2 \frac{sin(a)}{a}$ where $a > 0$ .

Anyone know how to do this one? Thanks!

-Dan

Perhaps the following might help:

$<br /> \sin (\theta)\ e^{ia \cos( \theta)} =-\frac{1}{ia}\ \frac{d}{d\theta} e^{ia \cos( \theta)}<br />$

RonL
November 16th 2006, 03:18 AM
topsquark

Quote:

Originally Posted by CaptainBlack

Perhaps the following might help:

$<br /> \sin (\theta)\ e^{ia \cos( \theta)} =-\frac{1}{ia}\ \frac{d}{d\theta} e^{ia \cos( \theta)}<br />$

RonL

I don't mind that you pointed out something I should have seen almost immediately. I don't mind that I figured this out late last night and didn't get to the post early enough to "save face" by mentioning I figured it out.

What I mind is that I spent hours working on a problem that I should have been able to answer in minutes. :mad:

Ah well. The learning process! :) Thanks CaptainBlack.

-Dan