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Dansuer
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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:
Last edited: