Hermite Polynomials

**iLikeMaths** · December 13th 2008, 04:03 PM

how do I calculate the first three Hermite Polynomials ?

thank you

**mr fantastic** · December 13th 2008, 04:08 PM

Originally Posted by iLikeMaths

how do I calculate the first three Hermite Polynomials ?

thank you

Start by reading the following:

Hermite polynomials - Wikipedia, the free encyclopedia

Hermite polynomials: Definition from Answers.com

http://kiwi.atmos.colostate.edu/grou...rmitePolys.pdf

Hermite Polynomial

Hermite polynomials - ALGLIB

**iLikeMaths** · December 13th 2008, 04:15 PM

thanks for the link, so the polynomials are

1
2x
4x^2 - 2 etc

I know that they are orthogonal but how do I prove/verify that they are?

Thanks

**mr fantastic** · December 13th 2008, 06:12 PM

Originally Posted by iLikeMaths

thanks for the link, so the polynomials are

1
2x
4x^2 - 2 etc

I know that they are orthogonal but how do I prove/verify that they are?

Thanks

Read the links carefully, do a little more research and see if you can find a proof. If you turn up nothing I'll post a proof in due course (but I'm betting you'll turn up something).

**iLikeMaths** · December 13th 2008, 07:03 PM

ive got to the point where i am integrating
1 and 2x and the weighting function e^x^-2 (sori i dont know how to use LaTex) between the intervals infinity and minus infinity but i dont understand how the integral equals to zero if the intervals do not have a value. i have integrated the polynomials and i got -e^x^-2, hope you understand thanks

**mr fantastic** · December 13th 2008, 07:39 PM

Originally Posted by iLikeMaths

ive got to the point where i am integrating
1 and 2x and the weighting function e^x^-2 (sori i dont know how to use LaTex) between the intervals infinity and minus infinity but i dont understand how the integral equals to zero if the intervals do not have a value. i have integrated the polynomials and i got -e^x^-2, hope you understand thanks

To solve $\int_{-\infty}^{+\infty} (1) (2x) e^{-x^2} \, dx$ make the substitution $u = -x^2$ and deal with the improper integral in the usual way (have you been taught about improper integrals and how to treat them?).

**Moo** · December 14th 2008, 12:11 AM

Originally Posted by mr fantastic

To solve $\int_{-\infty}^{+\infty} (1) (2x) e^{-x^2} \, dx$ make the substitution $u = -x^2$ and deal with the improper integral in the usual way (have you been taught about improper integrals and how to treat them?).

Or just note that the integrand $2xe^{-x^2}$ is an odd function, and hence :
$\int_0^\infty 2xe^{-x^2} ~ dx=-\int_{-\infty}^0 2xe^{-x^2} ~ dx$

Since $\int_{-\infty}^{\infty}=\int_0^\infty+\int_{-\infty}^0$ , the integral is 0.

**iLikeMaths** · December 14th 2008, 06:26 AM

i still dont understand , can i post the actual question because i think i am not explaining the question properly

**mr fantastic** · December 14th 2008, 02:03 PM

Originally Posted by iLikeMaths

i still dont understand , can i post the actual question because i think i am not explaining the question properly

What do you not understand in post #6?

Please feel free to post the actual question.

It looks to me like you will need to revise and consolidate some of the background mathematics that your question expects you to understand and use.

**iLikeMaths** · December 14th 2008, 02:19 PM

[LEFT]The Actual Question

Using the definition of orthogonality, verify that the polynomials found in 1.(i.e. $1,2x, 4x^2 -2$ [FONT=CMR12])are mutually
orthogonal. You may use the following integrals without proof
$\int_{-\infty}^{\infty}e^{-x^2}~dx = \sqrt\pi$

i think my actual problem is that i missed the class on this topic and i am totally confused, the book i am using is showing me a different way to do it, if you could please explain in a very simple way, thanks

**mr fantastic** · December 14th 2008, 02:27 PM

Originally Posted by iLikeMaths

The Actual Question

Using the definition of orthogonality, verify that the polynomials found in 1.(i.e. $1,2x, 4x^2 -2$ )are mutually
orthogonal. You may use the following integrals without proof

Orthogonal with respect to

1. The weighting function $e^{-x^2}$

2. over the interval $-\infty < x < + \infty$

???

One of the required calculations has already been shown to you.

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