Ilya Zakharevich on Fri, 12 May 2000 13:51:09 -0400

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Re: L-series derivatives

On Fri, May 12, 2000 at 02:20:14PM +0100, Tom Womack wrote:
> Is there any particular reason why computing L(s) for an elliptic curve is
> an in-built command, whilst to compute L' you have to do something like

A complete support for a transcendental function M(x) in PARI should
contain the following algorithms:

  a) an algo to calculate all the derivatives of M(a) at any point a;

  b) for a polynomial P(x) an algorithm to calculate a polynomial Q(x)
     of degree deg(P)-1 such that M(x)-Q(x) is divisible by P(x);

"a" is needed to calculate values of M(a + x + O(x^100)) anyway, so
why not make a primitive which would allow one to access all the
derivatives of the builtin function?  For this we will need to extend
the struct which keeps the mapping from function names to C functions
to allow putting entries for derivatives.  What to do with functions
of several arguments needs discussion.

"b" is needed to calculate M(m), where m is a matrix.  Currently no
transcendental function supports it (!), which is probably one of the
reasons for the silly semantic of calculating M(m) component-wise.  ;-)