Math Is Fun Forum

dannyv

Member

Registered: 2007-09-20

Posts: 34

a very tricky integral

Hi, can someone give me a hand with this little integral please.

where
t is the time, which is discrete
s is between 0 and t
k has domain [-pi, pi]
n is a natural number
a, b are constants

actually this integral is the binomial expansion of this other integral

there are some other constants, which I omitted. Also, I am omitting the coefficients and summation of the binomial expansion in the first integral.

I looking for a closed form, and I tried to use asymptotic approximations but it doesn't work because in general to solve it using asymptotic techniques you need to write in this form:

and I can't do that. Maybe there are some other asymptotic techniques that I don't know, maybe you can give me a hand on this.

dannyv

Member

Registered: 2007-09-20

Posts: 34

Re: a very tricky integral

Maybe Laplaces's method could work. It requires an integral with the form:

Take a maximum point

of the function

, then the integral is:

I think that this method only requires

and

to be analytic.

dannyv

Member

Registered: 2007-09-20

Posts: 34

Re: a very tricky integral

for a good reference on asymptotics, this book seems really nice

http://books.google.co.jp/books?id=KQHPHPZs8k4C&printsec=frontcover#v=onepage&q=&f=false

and you can read it a lot for free from google books!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: a very tricky integral

Hi dannyv;

Thanks for the link of the book. I would rule out a closed form for that. Did you try IBP, it is the usual way to get an asymptotic form for an integral.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

dannyv

Member

Registered: 2007-09-20

Posts: 34

Re: a very tricky integral

what's IBP?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: a very tricky integral

Hi dannyv;

Sorry, meant integration by parts.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

dannyv

Member

Registered: 2007-09-20

Posts: 34

Re: a very tricky integral

ahhh, I see thx. Anyway, I tried it, but the integral just gets worst and worst, but according to other people, Laplaces's method seems to be the way here. Thanks for the help!!