# Remembering Special Function Equations...

I have an awful memory when it comes to factoids, I need to remember the Legendre, Hermite, Laguerre, Chebyshev, Hypergeometric & Jacobi equations, all of which are of the form $p(x)y'' + q(x)y' + r(x)y = 0$, where $p$ is a second degree polynomial, $q$ is a first degree polynomial & $r$ is a zero'th degree polynomial (interpreted as an eigenvalue).

Now, I can derive $r(x)$ by following Arfken's development & just substituting in a series solution & deriving what the eigenvalue should be.

Thus I'm left with finding out a way to remember the coefficients $p(x)$ & $q(x)$ for the Legendre, Hermite, Laguerre, Chebyshev, Hypergeometric & Jacobi equations. Is there any simple way to do this? Any unifying procedure or thought process?

Considering that this question could have been asked by including the $r(x)$ term, would you have recommended subbing in the series to derive the eigenvalue, or just told me to remember them all? :tongue:

I won't start asking about ways to remember things like Bessel's equation, Weber, Matthieu, Lame etc... as I feel that is a fruitless task, though any nice ideas or tips on these monstrosities would be greatly appreciated

memorization of anything usually calls for brute force. Just write equations with labels and names and study it like you would a vocab list from middle school.

Hi bolbteppa !

I fully agree with you. Actually, the choice of the names of the special functions is not following any logical way. So, memorization is a matter of "brute force" (as said by X89codered89X). The good feeling of being able to rely on Handbooks of Special Functions !
I would like to quote a sentence (translated from French) :
<< Certainly, the habit is praiseworthy and is full of good intentions to give mathematicians' names to the functions. It marks the gratitude to these Great Men. It is a way of making them immortal, even more than to be Academicians ! (On the French very honorific sense of this name). But, in return, where is the logic of classification and hierarchical organization? Obviously, it can be made independently of the awarded names. But then, so much efforts of memory, so much time spent in researches more historic and patronymic than mathematics! ... >>
From the paper: "Safari in the Contry of Special Functions", page 28.
http://www.scribd.com/JJacquelin/documents

HEY bolbteppa !
I am undergraduate student in physics and wondering to get answer of same question. It's been 3 years so if you got answer to memorizing Special functions(especially Bassel function) so let me know,Please.

Hi Jimit !
Unfortunately, nothing change. Special functions remain a matter of brute force for the memory.
As it is written as a joke at the end of the paper quoted above : << When the
best-seller "Phylogenetic Classification of the Special Functions" will appear ? >> .
There is no indication that there is any light at the end of the tunnel.

jim mcnamara
I need to remember the Legendre, Hermite, Laguerre, Chebyshev, Hypergeometric & Jacobi equations

Why?

The only reason I could imagine needing to memorize all these equations is for a test. I can't imagine a subject where you'd need to know all these equations.
Even if there was, it's unreasonable for a professor to expect you to know all the equations for the common special function.

There are a number of useful references, like Abramowitz and Stegun, that you can use to look up special function. You're much better off buying one of these references and learning how to use it efficiently.