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alfredska
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Expanding a function in "Gaussian-Hermites"
This isn't homework or coursework, but seeing as it's most like a homework problem, I figured this would be the best place to ask.
Note that I'm using the physics
I would like to expand a function (let's take a gaussian for example) in terms of this series (very similar to the harmonic oscillator, except for a factor of 2 in the exponential):
[tex]s\left(x\right)=\sum_n\alpha_nH_n\left(x\right)\exp\left(-x^2\right)[/tex]
To find the coefficients:
[tex]\alpha_n=\int_{-\infty}^{\infty}F\left(x\right)H_n\left(x\right)\exp\left(-x^2\right)dx/Normalization[/tex]
where [tex]F\left(x\right)=\exp\left(-\frac{x^2}{\sigma^2}\right)[/tex] in my example
I have figured out the normalization of [tex]H_n\left(x\right)\exp\left(-x^2\right)[/tex]
[tex]\alpha_n=\int_{-\infty}^{\infty}\left[H_n\left(x\right)\exp\left(-x^2\right)\right]^2dx=\sqrt{\frac{\pi}{2}}\left(2 n-1\right)![/tex]
but apparently I'm doing something wrong when I write:
[tex]\alpha_n=\frac{\int_{-\infty}^{\infty}\exp\left(-\frac{x^2}{\sigma^2}\right)H_n\left(x\right)\exp\left(-x^2\right)dx}{\sqrt{\sqrt{\frac{\pi}{2}}\left(2 n-1\right)!}}[/tex]
Can you tell me where I'm making my mistake?
This isn't homework or coursework, but seeing as it's most like a homework problem, I figured this would be the best place to ask.
Note that I'm using the physics
Homework Statement
I would like to expand a function (let's take a gaussian for example) in terms of this series (very similar to the harmonic oscillator, except for a factor of 2 in the exponential):
[tex]s\left(x\right)=\sum_n\alpha_nH_n\left(x\right)\exp\left(-x^2\right)[/tex]
Homework Equations
To find the coefficients:
[tex]\alpha_n=\int_{-\infty}^{\infty}F\left(x\right)H_n\left(x\right)\exp\left(-x^2\right)dx/Normalization[/tex]
where [tex]F\left(x\right)=\exp\left(-\frac{x^2}{\sigma^2}\right)[/tex] in my example
The Attempt at a Solution
I have figured out the normalization of [tex]H_n\left(x\right)\exp\left(-x^2\right)[/tex]
[tex]\alpha_n=\int_{-\infty}^{\infty}\left[H_n\left(x\right)\exp\left(-x^2\right)\right]^2dx=\sqrt{\frac{\pi}{2}}\left(2 n-1\right)![/tex]
but apparently I'm doing something wrong when I write:
[tex]\alpha_n=\frac{\int_{-\infty}^{\infty}\exp\left(-\frac{x^2}{\sigma^2}\right)H_n\left(x\right)\exp\left(-x^2\right)dx}{\sqrt{\sqrt{\frac{\pi}{2}}\left(2 n-1\right)!}}[/tex]
Can you tell me where I'm making my mistake?
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