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Adrien-Marie Legendre

lived from 1752 to 1833

Legendre's major work on elliptic integrals provided basic analytical tools for mathematical physics. He gave a simple proof that pi is irrational as well as the first proof that pi sqr is irrational.


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http://primes.utm.edu/glossary/page.php?sort=LegendreSymbol

http://hyperphysics.phy-astr.gsu.edu/hbase/math/legend.html

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--Last edited by saucer on 2007-01-02 19:34:23 --

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http://web.uconn.edu/~cdavid/latex2html/thermo2/thermo2.html


http://thesaurus.maths.org/mmkb/entry.html?action=entryById&id=3426


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--Last edited by saucer on 2007-08-10 20:27:01 --

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Legendre and other Math


http://www.du.edu/~jcalvert/math/mathom.htm


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citing

..If X is a vector, then P is an (n+1)-by-q matrix, where q = length(X). Each element P(m+1,i) corresponds to the associated Legendre function of degree n and order m evaluated at X(i).

In general, the returned array P has one more dimension than X, and each element P(m+1,i,j,k,...) contains the associated Legendre function of degree n and order m evaluated at X(i,j,k,...). Note that the first row of P is the Legendre polynomial evaluated at X, i.e., the case where m = 0...





http://www.mathworks.com/access/helpdesk/help/techdoc/ref/index.html?/access/helpdesk/help/techdoc/ref/legendre.html&http://www.google.co.uk/search?hl=en&q=Legendre+functions&btnG=Google+Search&meta=


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Special functions

http://gershwin.ens.fr/vdaniel/Doc-Locale/Cours-Mirrored/Methodes-Maths/white/math/s8/s8legd/s8legd.html


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--Last edited by saucer on 2007-01-11 05:02:21 --

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Legendre math links

http://encyclopedia.thefreedictionary.com/legendre

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Hi there,

for computing Bessel functions and Legendre polynomials there exists
recursive formulas (see for example Numerical Recipes in F77, p.172),
which can be implemented straightforward using only a few lines of
Fortran. Comparing with iterative (and much more sophisticated)
versions, the recursive versions are *really* slow.

The example below computes the Legendre polynomials recursively as
given in equations (6.8.7), (6.8. and (6.8.9) from Numerical Recipes
in F77, p.247

Only for 'l' up to 20 this version is suitable, for higher orders it is
really slow and takes ages. The iterative version from Numerical
Recipes however has no problems at all. Similar problems are
encountered with the computation of the modified Bessel functions with
fractional order ...

So my question: why is this so slow compared with the iterative
versions. Is this an 'inherently' problem of recursion? Is it my
program?

.................

Example:

pure recursive function legendrePoly(x,m,l) result(Plm)

! Computes the (associated) Legendre polynomial Plm(x)
! with (0 <= m <= l) and (-1 <= x <= 1) after the recursive
! formula in Numerical Recipes in Fortran 77 (Volume 1, p. 247).
! 'Ordinary' Legendre polynomals are given for m=0.
!
! Calls: pure recursive function ifact -> double factorial n!!

implicit none

real(dp) :: Plm
real(dp), intent(in) :: x
integer, intent(in) :: m
integer, intent(in) :: l

if(l == m) then
Plm = (-1)**m * ifact(2.0d0*m-1.0d0) * (1.0d0-x**2)**(m/2.0d0)
elseif(l == m-1) then
Plm = 0.0d0
else
Plm = (x*(2.0d0*l-1.0d0)*legendrePoly_(x,m,l-1) - &
(l+m-1.0d0)*legendrePoly_(x,m,l-2)) / (l-m)
endif
end function legendrePoly  

--Last edited by zee on 2007-01-27 00:26:00 --

I'm looking for the simplest elementary arithmetical proof, which
doesn't use legendre symbol, that

Let p, a prime number
2^((p-1)/2) = 1 (mod p) => p = ±1 (mod 8)

Could you give me a tip

Thank you in advance

See Gauss's Disquisitiones Arithmeticae.





I claim first that if m = +- 3 (mod  then x^2 = 2 (mod m) is insoluble.
Suppose false and m is the smallest natual number with x^2 = 2 (mod m)
soluble. Then there is an odd a with 0 < a < m and a^2 = 2 (mod m).
That is, a^2 - 2 = mn. As a^2 - 2 equiv 7 (mod  then n = -+3 (mod  
too. We can't have a = 1, and so 0 < mn < m^2 so n < m. As a^2 = 2
(mod n) then we have a contradiction.

As in standard theory, if 2^{(p-1)/2} = 1 (mod p) then 2 is a
square modulo p. (The equation x^{(p-1)/2} = 1 (mod p) has <= (p-1)/2
solutions but they include 1^2, 2^2, ...., ((p-1)/2)^2).

noting that where is the gamma function for integers , the definition can be generalized to complex values


http://mathworld.wolfram.com/Factorial.html

zee   -


Do you think Legendre's work on elliptic integrals is complicated to relate to for the non-scientist. I am reading a paper of rational functions, in this area.  

--Last edited by very very small tic on 2007-02-05 10:49:49 --

Quote :

very very small tic wrote : zee   - Do you think Legendre's work on elliptic integrals is difficult to relate to for the non-scientist. I am reading a paper of rational functions, in this area.

viz -
elliptic integrals cannot be expressed in terms of elementary functions; exceptions to this are when P does have repeated roots, or when R(x,y) contains no odd powers of y. zee  

--Last edited by zee on 2007-02-05 10:51:17 --

Orthogonals

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http://modular.math.washington.edu/SAGE/doc/html/ref/module-sage.functions.orthogonal-polys.html



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short history .. P S Laplace, too

http://library.thinkquest.org/22584/temh3017.htm



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--Last edited by saucer on 2007-03-07 10:37:37 --

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http://fermatslasttheorem.blogspot.com/2005/05/fermats-one-proof.html


saucer




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--Last edited by saucer on 2007-04-20 23:08:38 --

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http://www.efunda.com/math/legendre/index.cfm


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Legendre's Prime Number Conjecture
Most historical accounts of the Prime Number Theorem mention Legendre's
experimental conjecture (made in 1798 and again in 1808) that

                                  x
                pi(x)  =   ---------------
                            log(x) - A(x)

where pi(x) is the number of primes less than x, and the limit of
A(x) as x goes to infinity is 1.08366....  In 1850, Tschebycheff
proved that Legendre's conjecture cannot be true unless 1.08366...
is replaced by 1.

Aside from the comment that Legendre's conjecture was based on
"experimental evidence", I've never seen an explanation of how he
actually arrived at the number 1.08366...  Here's a table giving
the actual values of A(x) for several values of x:

                       x        A(x)
                     -----     ------
                     10^2      0.6052
                     10^3      0.9553
                     10^4      1.0736
                     10^5      1.0876
                     10^6      1.0763
                     10^7      1.0710
                     10^8      1.0639
                     10^9      1.0566
                     10^10     1.0504
 
I've never been able to see how anyone could infer a limit of 1.08366
from this table, or even from any truncated version of this table
(allowing for the possibility that Legendre may not have had the
values of pi(x) for very large values of x).  Notice that he gave
the "constant" to five significant digits, which seems remarkable
working from this kind of data.

This raises some questions:
(1) How did Legendre arrive at the constant 1.08366...?
(2) For what precise value of x does A(x) achieve it's maximum value?

Regarding (1), I wonder if there is any connection with the limit of

                    1         /     1 \
                   ---  PROD ( 1 + --- )  
                  ln(x)       \     p /

where the product is evaluated over all primes p < x.  I believe that
the infinite product is known to equal

              6 e^(gamma)
              -----------  =~  1.082762...
                 pi^2

which is fairly close to Legendre's constant 1.08366...  Is it possible
that Legendre was aware of this infinite product (or some estimate of
it) when he made his Prime Number conjecture?

I don't have an explicit reference for the above evaluation of the
infinite product, but it follows closely from a combination of Theorem
302 in Hardy & Wright's "Introduction to the Theory of Numbers"

              zeta(s)
              -------  =    PROD (1 + p^-s)           (s>1)
              zeta(2s)

and "a formula of Mertens" given on page 162 of Ribenboim's "Book of
Prime Number Records"
                                1                1
        e^gamma   =   lim    -------   PROD  ----------
                     n->inf   log(n)   i < n   1 - 1/p_i

along with the fact that zeta(2)=pi^2/6.

The only other information I've found is in Tchebyshev's paper where
he says Legendre

       "..begins by comparing his formula with the result of
        counting the primes in the most extended tables,
        namely those from 10,000 up to 1,000,000, after
        which he applies his formula to the solution of many
        problems".

This doesn't clear up the mystery for me, because by 1,000,000 the
value A(x) has already passed its maximum and is down to 1.076...  
So I still don't see how  Legendre arrived at the precise value
1.08366...

Regarding my question (2), which asked for the maximum value of A(x),
I've computed A(p) for p < 10^6 and found that the maximum value is
1.1119625..., occurring at p = 24137, which is the 2688th prime.




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Legendre's, Chebyshev's, Hermite's and Jacobi's orthogonal polynomials evaluated using reccurence formulae. ( This calculator is un-tested, by me ....)


http://www.matf.bg.ac.yu/r3nm/NumericalMethods/Integration/OrthogonalPolynomial2.html  

--Last edited by saucer on 2007-08-10 03:39:52 --

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http://www.uconn.edu/~cdavid/latex2html/thermo3/thermo3.html



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