Talk:Frobenius solution to the hypergeometric equation

Response
Thanks for your advice, I included an introduction in which I describe the differential equation, why it is useful to solve it, and what I will be doing.

I hope this meets your criteria. It should be noted that this article is mainly for mathematicians. But I object deleting it as it contains much useful information. And I did include an introduction for the "casual reader".

To knowledge :) --MostafaSabry (talk) 20:29, 19 March 2008 (UTC)

Moved to Wikibooks
This page was moved to Frobenius solution to the hypergeometric equation. I think it is better to keep the work there and delete this page under WP:SD. Carlosguitar (ready and willing) 08:08, 21 March 2008 (UTC)


 * The transwiki'd version omits the partial reformatting we have made here, though. If it does get deleted, that had better be moved first. Ryan Reich (talk) 19:13, 21 March 2008 (UTC)


 * Note also that PlanetMath might be a good place for this. They like all-tex markup, and have a policy that explicitly allows proofs such as this. linas (talk) 02:36, 27 March 2008 (UTC)

The reference
is not specified (or spelt correctly) or sufficiently canonical. Better references are needed. Xxanthippe (talk) 08:12, 21 March 2008 (UTC).
 * Dr Sayed Abed. Ordinary Differntial Equations.

I know, the idea is that the book was published in my university, so that it doesn't have an ISBN...etc. I just checked the method of Frobenius there, so maybe I can find another reference. --MostafaSabry (talk) 09:51, 22 March 2008 (UTC)


 * ? Are you sure? As a teen, I worked for a tiny publisher who was always careful to get ISBN's for their books, it's hard to believe a university press wouldn't take such a simple step. linas (talk) 02:34, 27 March 2008 (UTC)

Finished conversion
The article has now been completely converted to standard Wiki markup; in some places, the LaTeX is still bad, but the article is now editable. I suggest that the Wikibooks version be replaced with this one, and that some decision as to the fate of this article be made. I don't see how it can be made encyclopedic, but it can probably be pointed to from hypergeometric equation and Frobenius series. Ryan Reich (talk) 19:26, 29 March 2008 (UTC)
 * I did the replacement myself. I have no idea how things work at Wikibooks, but the text is there now. Ryan Reich (talk) 20:35, 2 April 2008 (UTC)

Comparison with Abramowitz and Stegun
The question I would like to ask is this.

In the expansions about z=0 the solutions to the case \gamma is an integer not equal to one ($\gamma=1-m$ where $m=1,2,...$) and ($\gamma=1+m$ where $m=1,2,3$...) does not include the finite sums seen in Abramowitz and Stegun (Editors) Chapter 15 (Fritz Oberhettinger) 15.5.20 and 15.5.21.

(Abramowitz and Stegun "Handbook of Mathematical Functions" Dover Books on Advanced Mathematics ISBN 0-486-61272-4.)

For instance, for c=1-m 15.5.21 has a term \sum_{n=1}^m \frac{ (n-1)! (-m)_n}{(1-a-m)_n(1-b-m)_n} which I have derived myself for and m=3 for instance. Can the author point out where I have misunderstood, or tell me whether this problem is real?

It seems to me that there is a mistake in the solution. In the case where $\gamma$ is negative or zero so that the $c=0$ root of the indicial equation gives an infinite coefficient. However, we must evaluate $\frac{\partial y_b}{\partial c}$ at the $c=0$ root as well. We do not evaluate this derivative at $c=1-\gamma$ as the solution u8nder discussion suggests. (See for instance Piaggio p114 Case III.)

On evaluating $\frac{\partial y_b}{\partial c}$ at $c=0$ I get same result as described in Abramowitz and Stegun which I think is the only way of obtaining that finite sum I mentioned before. Similarly, if $c>2,3,4...=1+m$ then the $c=1=\gamma$ root throws up an infinity. In this case we must evaluate $\frac{\partial y_b}{\partial c}$ at $c=1-\gamma$ and not $c=0$. Using $c=1-\gamma$ in the partial derivative yields a finite $m$ terms (over r say) in $x^{-r}$ as in Abramowitz and Stegun 15.5.19. This finite sum is not in the article under discussion.

So, since my original question, a have convinced myself that this part of the article is indeed incorrectTethys sea (talk) 19:07, 12 February 2016 (UTC)

— Preceding unsigned comment added by Tethys sea (talk • contribs) 20:09, 10 February 2016 (UTC)

UPDATE....

These two sections (c=0,-1,2 and c=2,3,4) should be deleted altogether.

They are contain the following egregious error. For the root c=0 they evaluate the partial derivative of y w.r.t. c at c=1-gamma. The partial derivative of the indicial polynomial is not zero at c=1-gamma, so this function in NOT A SOLUTION of the hypergeometric equation. For c=1-gamma they evaluate the partial derivative at c=0, same problem. This is why there is no way of getting them into the forms of the standard solution. The given solutions are pre garbage! It's not just a matter of a few missing terms... 86.183.91.156 (talk) 16:20, 7 April 2016 (UTC)

I have re-written section 3.3
As I mentioned previously, the results in section 3.3 (with sub-sections gamma an integer <=0 and gamma >1) were wrong since y_2 was derived by evaluating the partial derivative at the wrong root of the indicial equation.

I have now replaced section 3.3 completely, starting with gamma 0 or negative I used the usual "replace a_0 with b_0 c" way to derive the solution. However, this is not the standard solution as supplied (for instance) by Abramowitz and Stegun. I have therefore included a section describing (one of the ways) in which the standard solution can be arrived at. I have finished things off with writing down the standard solution for the case gamma >1. I would be thankful for corrections regarding typos and html style (I am not used to html at all) and so on. Tethys sea (talk) 14:28, 30 May 2016 (UTC) — Preceding unsigned comment added by Tethys sea (talk • contribs) 14:18, 30 May 2016 (UTC)