Show that

Printable View

- Aug 14th 2009, 11:14 AMluoboIntegral
Show that

- Aug 14th 2009, 11:24 AMMatt Westwood
I'd try substituting and see where that got me. What have you tried so far?

Sorry, ignore me - I didn't notice this was a challenge forum! Okay, I'll have a go ... - Aug 14th 2009, 12:22 PMMatt Westwood
Hang on, this is Bessel functions, isn't it? Way over my head, sorry about that. I'm going to have to give this one a miss till I've learned up on it.

- Aug 14th 2009, 12:30 PMTwistedOne151
First, use the sum of angles formula on :

,

so our integral becomes:

Now, one can simply note that , and that , so that our integral is

We recogize this as having the form , and so:

-Kevin C. - Aug 14th 2009, 12:47 PMNonCommAlg
- Aug 14th 2009, 01:03 PMluobo
That's right. I just happened to find out this interesting integral when I was doing the line integral posted here today by transgalactic, using both line integral and double integral.

http://www.mathhelpforum.com/math-he...-question.html

Now you have got this: , can you go ahead and find

- Aug 14th 2009, 01:08 PMNonCommAlg
- Aug 14th 2009, 01:19 PMluobo
To me, that's the beauty of this integral. You can continue the differentiation to find and you can also find the curve is a beauty as well in when you plot it out.

I am sure NonCommAlg has a solution for , so can you please share with us?

**Additional question is: What happens if is zero or negative. Is there a closed-form solution. I guess not, if so, how about if . I don't have a solution for this. If you have, can you please share.** - Aug 14th 2009, 03:37 PMluobo
- Aug 14th 2009, 06:50 PMluobo
(*)

[/COLOR][/COLOR] (1)__Let me give an example showing how the previous equality is used to get the summation of an important series.__

According to the equality,

(2)

Therefore,

(3)

Taylor's series,

(4)

Plug (3) into (4),

(5)

Multiply on both sides,

(6)

**Finally, the series:**

(7)

**Some Special cases:**

(a)

(8)

(b)

(9)

(c)

(10) (which is within everybody's expectation)

By Luobo

08/14/2009 - Aug 14th 2009, 06:53 PMpankaj