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Find the first three coeficents c_n of the expansion of Cos(x) in Hermite Polynomials.

The first three Hermite Polinomials are:

[itex]H_0(x) = 1[/itex]

[itex]H_1(x) = 2x[/itex]

[itex]H_0(x) = 4x^2-2[/itex]

I know how to solve a similar problem where the function is a polynomial of finite degree, say x^3. Using the fact that H_n(x) is a polinomial of degree n, i set all the coeficent after c_3 equal to zero and equate the terms with equal degree. I find a system of linear equations and i solve it.

In this case however the taylor series of Cos(x) is a polinomial of infinite degree. I can't apply this method.

I then try to use the orthogonality of the Hermite polynomials

[itex]\int^{∞}_{-∞}e^{-x^2}H_n(x)H_m(x)dx = \sqrt{\pi}2^nn!\delta_{nm}[/itex]

From the orthogonality i find the coeficents to be

[itex]c_0 = \frac{1}{\sqrt{\pi}2^nn!}\int^{∞}_{-∞}e^{-x^2} Cos(x) dx[/itex]

[itex]c_1 = \frac{1}{\sqrt{\pi}2^nn!}\int^{∞}_{-∞}e^{-x^2} 2xCos(x) dx[/itex]

[itex]c_2 = \frac{1}{\sqrt{\pi}2^nn!}\int^{∞}_{-∞}e^{-x^2} (4x^2-2)Cos(x) dx[/itex]

Which are three hard integrals i haven't been able to solve. I can't use computer methods as i'm suppose to solve this in an exam with pen and paper. I'm stuck.

Thanks to anyone who takes a look at this.:tongue2:

## Homework Statement

Find the first three coeficents c_n of the expansion of Cos(x) in Hermite Polynomials.

The first three Hermite Polinomials are:

[itex]H_0(x) = 1[/itex]

[itex]H_1(x) = 2x[/itex]

[itex]H_0(x) = 4x^2-2[/itex]

## The Attempt at a Solution

I know how to solve a similar problem where the function is a polynomial of finite degree, say x^3. Using the fact that H_n(x) is a polinomial of degree n, i set all the coeficent after c_3 equal to zero and equate the terms with equal degree. I find a system of linear equations and i solve it.

In this case however the taylor series of Cos(x) is a polinomial of infinite degree. I can't apply this method.

I then try to use the orthogonality of the Hermite polynomials

[itex]\int^{∞}_{-∞}e^{-x^2}H_n(x)H_m(x)dx = \sqrt{\pi}2^nn!\delta_{nm}[/itex]

From the orthogonality i find the coeficents to be

[itex]c_0 = \frac{1}{\sqrt{\pi}2^nn!}\int^{∞}_{-∞}e^{-x^2} Cos(x) dx[/itex]

[itex]c_1 = \frac{1}{\sqrt{\pi}2^nn!}\int^{∞}_{-∞}e^{-x^2} 2xCos(x) dx[/itex]

[itex]c_2 = \frac{1}{\sqrt{\pi}2^nn!}\int^{∞}_{-∞}e^{-x^2} (4x^2-2)Cos(x) dx[/itex]

Which are three hard integrals i haven't been able to solve. I can't use computer methods as i'm suppose to solve this in an exam with pen and paper. I'm stuck.

Thanks to anyone who takes a look at this.:tongue2:

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