Integrated wavefunction

Hi,

I've been working through my textbook and come across a method of integration that I'm not familiar with.

In the attached file the improper integral produces a function that includes factorials and other deviations from normal single and multi-variable integration that I am accustomed to.

Could someone identify which branch of maths this it? i.e. Is it the Gamma function?

Thanks

WP_20150608_001.jpg413.2KB

Reply 1

mik1a

10

Yes,

See the first paragraph of

http://www.math.toronto.edu/lgoldmak/Feynman.pdf

"Differentiating under the integral sign" - a pretty neat trick that is relatively unknown amongst undergrad students.

Reply 2

poorform

10

Original post by mik1a

Yes,

See the first paragraph of

http://www.math.toronto.edu/lgoldmak/Feynman.pdf

"Differentiating under the integral sign" - a pretty neat trick that is relatively unknown amongst undergrad students.

Interesting read.

Reply 3

a nice man

OP

9

Original post by mik1a

Yes,

See the first paragraph of

http://www.math.toronto.edu/lgoldmak/Feynman.pdf

"Differentiating under the integral sign" - a pretty neat trick that is relatively unknown amongst undergrad students.

Very interesting, thanks for the help!

Reply 4

atsruser

11

Original post by a nice man

Could someone identify which branch of maths this it? i.e. Is it the Gamma function?

You don't need anything as sophisticated as DUTIS or gamma functions. We have, via IBP:

I_n = \int_0^\infty x^n e^{-ax}\ dx = -\frac{1}{a} [x^ne^{-ax}]_0^\infty + \frac{n}{a}\int_0^\infty x^{n-1}e^{-ax}\ dx = \frac{n}{a}I_{n-1}

An easy integration shows that

I_0= \int_0^\infty e^{-ax}\ dx=\frac{1}{a}

so we get:

I_0=\frac{1}{a}

I_1=\frac{1}{a} \frac{1}{a} = \frac{1}{a^2}

I_2=\frac{2}{a}\frac{1}{a^2} = \frac{2!}{a^3}

I_3=\frac{3}{a}\frac{2!}{a^3} = \frac{3!}{a^4}

...

I_n=\frac{n!}{a^{n+1}}

This turns out nice because of the limits that you are integrating over. It's messier for a finite upper limit - the integral is then the lower incomplete gamma function, I think.

Integrated wavefunction

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