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substance90
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Homework Statement
Calculate the following integrals:
(a) [tex]I(n,\alpha) = \int_{0}^{\infty} e^{-\alpha x^2}x^n dx[/tex] for n whole integers and [tex]n \ge 0[/tex]
Calculate all results till n=5.
Tip: First calculate [tex]I^2(0,\alpha)[/tex] and [tex]I(1,\alpha)[/tex] and then use this to calculate n>1.
(b) [tex]I(n)=\int_{0}^{\infty} e^{-x}x^n dx[/tex] for n whole and half integer where [tex]n\ge -1/2[/tex]
Calculate all results till n=5
Tip: Calculate I(n) using I(0) and I(-1/2).
Homework Equations
The Attempt at a Solution
(а) I managed to do the first part using polar coordinates and substitution [tex]I(0,\alpha) = \sqrt{\frac{\pi}{\alpha}}[/tex] but I keep getting 0 for [tex]n \ge 1[/tex]
For example with 1: [tex]I(1,\alpha) = \int_0^{\infty} e^{-\alpha x^2} x^1 dx = x \sqrt{\frac{\pi}{\alpha}} -\int_0^{\infty} \sqrt{\frac{\pi}{\alpha}} 2x dx = x^2 \sqrt{\frac{\pi}{\alpha}} - x^2 \sqrt{\frac{\pi}{\alpha}} = 0[/tex]
(b) The first part here is ok: [tex]I(0) = \int_0^{\infty} e^{-x} dx = -e^{-x}[/tex] but I tried a thousand times to do I(-1/2) and keep going in circle with integration by parts.
Any ideas would be greatly appreciated!