(maxima.info)Functions and Variables for orthogonal polynomials
77.2 Functions and Variables for orthogonal polynomials
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-- Function: assoc_legendre_p (<n>, <m>, <x>)
The associated Legendre function of the first kind of degree <n>
and order <m>.
Reference: Abramowitz and Stegun, equations 22.5.37, page 779,
8.6.6 (second equation), page 334, and 8.2.5, page 333.
-- Function: assoc_legendre_q (<n>, <m>, <x>)
The associated Legendre function of the second kind of degree <n>
and order <m>.
Reference: Abramowitz and Stegun, equation 8.5.3 and 8.1.8.
-- Function: chebyshev_t (<n>, <x>)
The Chebyshev polynomial of the first kind of degree <n>.
Reference: Abramowitz and Stegun, equation 22.5.47, page 779.
-- Function: chebyshev_u (<n>, <x>)
The Chebyshev polynomial of the second kind of degree <n>.
Reference: Abramowitz and Stegun, equation 22.5.48, page 779.
-- Function: gen_laguerre (<n>, <a>, <x>)
The generalized Laguerre polynomial of degree <n>.
Reference: Abramowitz and Stegun, equation 22.5.54, page 780.
-- Function: hermite (<n>, <x>)
The Hermite polynomial of degree <n>.
Reference: Abramowitz and Stegun, equation 22.5.55, page 780.
-- Function: intervalp (<e>)
Return 'true' if the input is an interval and return false if it
isn't.
-- Function: jacobi_p (<n>, <a>, <b>, <x>)
The Jacobi polynomial.
The Jacobi polynomials are actually defined for all <a> and <b>;
however, the Jacobi polynomial weight '(1 - <x>)^<a> (1 + <x>)^<b>'
isn't integrable for '<a> <= -1' or '<b> <= -1'.
Reference: Abramowitz and Stegun, equation 22.5.42, page 779.
-- Function: laguerre (<n>, <x>)
The Laguerre polynomial of degree <n>.
Reference: Abramowitz and Stegun, equations 22.5.16 and 22.5.54,
page 780.
-- Function: legendre_p (<n>, <x>)
The Legendre polynomial of the first kind of degree <n>.
Reference: Abramowitz and Stegun, equations 22.5.50 and 22.5.51,
page 779.
-- Function: legendre_q (<n>, <x>)
The Legendre function of the second kind of degree <n>.
Reference: Abramowitz and Stegun, equations 8.5.3 and 8.1.8.
-- Function: orthopoly_recur (<f>, <args>)
Returns a recursion relation for the orthogonal function family <f>
with arguments <args>. The recursion is with respect to the
polynomial degree.
(%i1) orthopoly_recur (legendre_p, [n, x]);
(2 n + 1) P (x) x - n P (x)
n n - 1
(%o1) P (x) = -------------------------------
n + 1 n + 1
The second argument to 'orthopoly_recur' must be a list with the
correct number of arguments for the function <f>; if it isn't,
Maxima signals an error.
(%i1) orthopoly_recur (jacobi_p, [n, x]);
Function jacobi_p needs 4 arguments, instead it received 2
-- an error. Quitting. To debug this try debugmode(true);
Additionally, when <f> isn't the name of one of the families of
orthogonal polynomials, an error is signalled.
(%i1) orthopoly_recur (foo, [n, x]);
A recursion relation for foo isn't known to Maxima
-- an error. Quitting. To debug this try debugmode(true);
-- Variable: orthopoly_returns_intervals
Default value: 'true'
When 'orthopoly_returns_intervals' is 'true', floating point
results are returned in the form 'interval (<c>, <r>)', where <c>
is the center of an interval and <r> is its radius. The center can
be a complex number; in that case, the interval is a disk in the
complex plane.
-- Function: orthopoly_weight (<f>, <args>)
Returns a three element list; the first element is the formula of
the weight for the orthogonal polynomial family <f> with arguments
given by the list <args>; the second and third elements give the
lower and upper endpoints of the interval of orthogonality. For
example,
(%i1) w : orthopoly_weight (hermite, [n, x]);
2
- x
(%o1) [%e , - inf, inf]
(%i2) integrate(w[1]*hermite(3, x)*hermite(2, x), x, w[2], w[3]);
(%o2) 0
The main variable of <f> must be a symbol; if it isn't, Maxima
signals an error.
-- Function: pochhammer (<x>, <n>)
The Pochhammer symbol. For nonnegative integers <n> with '<n> <=
pochhammer_max_index', the expression 'pochhammer (<x>, <n>)'
evaluates to the product '<x> (<x> + 1) (<x> + 2) ... (<x> + n -
1)' when '<n> > 0' and to 1 when '<n> = 0'. For negative <n>,
'pochhammer (<x>, <n>)' is defined as '(-1)^<n> / pochhammer (1 -
<x>, -<n>)'. Thus
(%i1) pochhammer (x, 3);
(%o1) x (x + 1) (x + 2)
(%i2) pochhammer (x, -3);
1
(%o2) - -----------------------
(1 - x) (2 - x) (3 - x)
To convert a Pochhammer symbol into a quotient of gamma functions,
(see Abramowitz and Stegun, equation 6.1.22) use 'makegamma'; for
example
(%i1) makegamma (pochhammer (x, n));
gamma(x + n)
(%o1) ------------
gamma(x)
When <n> exceeds 'pochhammer_max_index' or when <n> is symbolic,
'pochhammer' returns a noun form.
(%i1) pochhammer (x, n);
(%o1) (x)
n
-- Variable: pochhammer_max_index
Default value: 100
'pochhammer (<n>, <x>)' expands to a product if and only if '<n> <=
pochhammer_max_index'.
Examples:
(%i1) pochhammer (x, 3), pochhammer_max_index : 3;
(%o1) x (x + 1) (x + 2)
(%i2) pochhammer (x, 4), pochhammer_max_index : 3;
(%o2) (x)
4
Reference: Abramowitz and Stegun, equation 6.1.16, page 256.
-- Function: spherical_bessel_j (<n>, <x>)
The spherical Bessel function of the first kind.
Reference: Abramowitz and Stegun, equations 10.1.8, page 437 and
10.1.15, page 439.
-- Function: spherical_bessel_y (<n>, <x>)
The spherical Bessel function of the second kind.
Reference: Abramowitz and Stegun, equations 10.1.9, page 437 and
10.1.15, page 439.
-- Function: spherical_hankel1 (<n>, <x>)
The spherical Hankel function of the first kind.
Reference: Abramowitz and Stegun, equation 10.1.36, page 439.
-- Function: spherical_hankel2 (<n>, <x>)
The spherical Hankel function of the second kind.
Reference: Abramowitz and Stegun, equation 10.1.17, page 439.
-- Function: spherical_harmonic (<n>, <m>, <x>, <y>)
The spherical harmonic function.
Reference: Merzbacher 9.64.
-- Function: unit_step (<x>)
The left-continuous unit step function; thus 'unit_step (<x>)'
vanishes for '<x> <= 0' and equals 1 for '<x> > 0'.
If you want a unit step function that takes on the value 1/2 at
zero, use '(1 + signum (<x>))/2'.
-- Function: ultraspherical (<n>, <a>, <x>)
The ultraspherical polynomial (also known as the Gegenbauer
polynomial).
Reference: Abramowitz and Stegun, equation 22.5.46, page 779.
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