Integrals and gamma functions manipulation

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1. Dec 29, 2014

mathsdespair

1. The problem statement, all variables and given/known data
I am working through some maths to deepen my understanding of a topic we have learnt about. However I am not sure what the author has done and I have copied below the chunk I am stuck on. I would be extremely grateful if someone could just briefly explain what is going on i.e how to get from one step to another and why.
Thanks

2. Relevant equations

$\int^\infty_y P(2y;2\nu,2k) dK$

where $P(2y;2\nu,2k)$ =$\frac{1}{2} (\frac{x}{\lambda})^{\frac{\nu-2}{4}} I_{\frac{\nu-2}{2}}(\lambda x)^{\frac{1}{2}} e^{\frac{-(\lambda+x)}{2}}$
Note that I is the bessel function of the first kind of order K and is defined as $(\frac{1}{2}Z) \sum\limits_{j=0}^\infty \frac{(\frac{z^2}{4})^j}{{j!\gamma(k+j+1)}}$
$$=\int^\infty_y e^{-z-k} (\frac{z}{k})^{\tau-1} (kz)^{\frac{\tau-1}{2}} \sum\limits_{n=0}^\infty \frac{(zk)^n}{n!\gamma(n+\nu-1+1)}dK$$
$$=\int^\infty_y \frac{e^{-z} z^{n+v-1}}{\gamma(n+\nu)} \int^\infty_y \frac{e^{-k} k^{n}}{\gamma(n+1)}dK$$
$$=\sum\limits_{n=0}^\infty g(n+v,z)G(n+1,y)$$

Please also note that $\gamma$ is meant to be that symbol that looks like a T when working with gamma funtions but I do not know what it is.

3. The attempt at a solution

The only thing I know is that it has something to do with gamma functions and integration by parts.
By the way this is not assignment related but something I really want to understand.

2. Dec 29, 2014

mathsdespair

Does anyone know why latex is not working on the first part of equation?

3. Dec 29, 2014

Ray Vickson

See remarks following the re-write of your post, which I have edited to:

Evaluate
$$J = \int_y^\infty P(2y;2\nu,2k) dk$$

where
$$P(2y;2\nu,2k) = \frac{1}{2} (\frac{x}{\lambda})^{\frac{\nu-2}{4}} I_{\frac{\nu-2}{2}}(\lambda x)^{\frac{1}{2}} e^{\frac{-(\lambda+x)}{2}} \leftarrow \text{makes no sense!}$$
Note that $I$ is the bessel function of the first kind of order $k$ and is defined as
$$\frac{z}{2} \sum_{j=0}^\infty \frac{(\frac{z^2}{4})^j}{{j!\gamma(k+j+1)}}$$
Thus, we have
$$J = \int_y^\infty e^{-z-k} (\frac{z}{k})^{\tau-1} (kz)^{\frac{\tau-1}{2}} \sum_{n=0}^\infty \frac{(zk)^n}{n!\gamma(n+\nu-1+1)}dk \\ \;\;=\int_y^\infty \frac{e^{-z} z^{n+v-1}}{\gamma(n+\nu)} \int_y^\infty \frac{e^{-k} k^{n}}{\gamma(n+1)}dk \\ \;\; =\sum_{n=0}^\infty g(n+v,z)G(n+1,y)$$

I did not understand what your \gamma is, so I have not addressed that issue.

Several points:
(1) Your definition of P makes no sense because you have y on one side and x on the other, and do not say what the relationship is between x and y.
(2) Best to avoid "\$" in LaTeX; just use "# # 'material' # # (remove blanks between the #s) to put 'material' in an in-line equation, or use [t e x] 'material' [/t e x] (remove spaces) to put it as a displayed equation.
(3) You can say "\sum_a^b"; no need to say "sum_limits_a^b", so that is what I have done in the above.
(4) Same as (2) but for integrals. It is customary (and good practice if you want to exchange documents with others) to say "int_a^b" rather than "int^b_a", so I changed those.
(5) Decide if you mean k or K and use it consistently; I changed dK to dk to fix it up.
(6) In a multi-line equation you can just do what I did above: start by using "[t e x]" (no spaces), use "\\" to end a line, then start the next line with an "=" or whatever. Keep going like that until you run out of lines, then end the thing by "[/ t e x]" (no spaces)
(7) The way you wrote it was confusing and misleading. You need a break after the definition of $I$ and before the remaining material. I did that by defining a symbol (J) for the thing you want and then saying "Thus, we have J = ..."

4. Dec 29, 2014

mathsdespair

Thanks for your reply and thanks for having a look. Gamma was meant to be $$\Gamma$$ (sorry about that)
I have copied what is above directly from the book.
Do you know how to get from j to the next step ?
Thanks

5. Dec 29, 2014

Ray Vickson

I cannot hope to make any headway on the problem until you tell me:
(1) In the definition of P, you have 2y on the left and x on the right. Is x = y, or what? Also, your definition of P has $\lambda$ in it, but there is no indication of how $\lambda$ is related to $\nu$ or $k$ or whatever.
(2) In the second-last line you have written two integrations (and no sum), but have only one integration variable "dk". Is that a typo? If not, what is the correct expression?

6. Dec 30, 2014

Staff: Mentor

Use pairs of dollar signs, not single ones.
$$\int^\infty_y P(2y;2\nu,2k) dk$$

7. Dec 30, 2014

mathsdespair

Hey mate its ok, I worked it out.
Thanks anyway