- #1
AdrianMay
- 121
- 4
Hi folks,
I'm trying to get from the established relation:
$$ \int_{-\infty}^{\infty} dx.x^2.e^{-\frac{1}{2}ax^2} = a^{-2}\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} $$
to the similarly derived:
$$ \int_{-\infty}^{\infty} dx.x^4.e^{-\frac{1}{2}ax^2} = 3a^{-4} \int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} $$
but instead I'm winding up with:
$$ \int_{-\infty}^{\infty} dx.x^4.e^{-\frac{1}{2}ax^2} = (4a^{-3} - a^{-4}) \int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} $$.
Evidently the difference is that I have ##a^{-3}## where I should have ##a^{-4}## but I can't seem to fault my own logic. First I differentiate the thing I started with:
$$ -2\frac{\partial}{\partial a} [ \int_{-\infty}^{\infty} dx.x^2.e^{-\frac{1}{2}ax^2} ] = -2\frac{\partial}{\partial a} [ a^{-2}\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} ] $$
apply the chain rule:
$$ \int_{-\infty}^{\infty} dx.x^4.e^{-\frac{1}{2}ax^2} = -2 \{ \frac{\partial a^{-2}}{\partial a}.\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} + a^{-2}.\frac{\partial}{\partial a}\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} \} $$
and hit the problem in what looks like the easy bit:
$$ \int_{-\infty}^{\infty} dx.x^4.e^{-\frac{1}{2}ax^2} = 4a^{-3}.\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} - a^{-4}.\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} $$
(where the last term follows from the relation I started with.)
So where's the bug?
Thanks in advance,
Adrian.
I'm trying to get from the established relation:
$$ \int_{-\infty}^{\infty} dx.x^2.e^{-\frac{1}{2}ax^2} = a^{-2}\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} $$
to the similarly derived:
$$ \int_{-\infty}^{\infty} dx.x^4.e^{-\frac{1}{2}ax^2} = 3a^{-4} \int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} $$
but instead I'm winding up with:
$$ \int_{-\infty}^{\infty} dx.x^4.e^{-\frac{1}{2}ax^2} = (4a^{-3} - a^{-4}) \int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} $$.
Evidently the difference is that I have ##a^{-3}## where I should have ##a^{-4}## but I can't seem to fault my own logic. First I differentiate the thing I started with:
$$ -2\frac{\partial}{\partial a} [ \int_{-\infty}^{\infty} dx.x^2.e^{-\frac{1}{2}ax^2} ] = -2\frac{\partial}{\partial a} [ a^{-2}\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} ] $$
apply the chain rule:
$$ \int_{-\infty}^{\infty} dx.x^4.e^{-\frac{1}{2}ax^2} = -2 \{ \frac{\partial a^{-2}}{\partial a}.\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} + a^{-2}.\frac{\partial}{\partial a}\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} \} $$
and hit the problem in what looks like the easy bit:
$$ \int_{-\infty}^{\infty} dx.x^4.e^{-\frac{1}{2}ax^2} = 4a^{-3}.\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} - a^{-4}.\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} $$
(where the last term follows from the relation I started with.)
So where's the bug?
Thanks in advance,
Adrian.
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