Previous: Introduction to orthogonal polynomials, Up: orthopoly [Contents][Index]
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.
‘Category: Package orthopoly’
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.
‘Category: Package orthopoly’
The Chebyshev polynomial of the first kind of degree n.
Reference: Abramowitz and Stegun, equation 22.5.47, page 779.
‘Category: Package orthopoly’
The Chebyshev polynomial of the second kind of degree n.
Reference: Abramowitz and Stegun, equation 22.5.48, page 779.
‘Category: Package orthopoly’
The generalized Laguerre polynomial of degree n.
Reference: Abramowitz and Stegun, equation 22.5.54, page 780.
‘Category: Package orthopoly’
The Hermite polynomial of degree n.
Reference: Abramowitz and Stegun, equation 22.5.55, page 780.
‘Category: Package orthopoly’
Return true
if the input is an interval and return false if it isn’t.
‘Category: Package orthopoly’ ‘Category: Predicate functions’
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.
‘Category: Package orthopoly’
The Laguerre polynomial of degree n.
Reference: Abramowitz and Stegun, equations 22.5.16 and 22.5.54, page 780.
‘Category: Package orthopoly’
The Legendre polynomial of the first kind of degree n.
Reference: Abramowitz and Stegun, equations 22.5.50 and 22.5.51, page 779.
‘Category: Package orthopoly’
The Legendre function of the second kind of degree n.
Reference: Abramowitz and Stegun, equations 8.5.3 and 8.1.8.
‘Category: Package orthopoly’
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);
‘Category: Package orthopoly’
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.
‘Category: Package orthopoly’
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.
‘Category: Package orthopoly’
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
‘Category: Package orthopoly’ ‘Category: Gamma and factorial functions’
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.
‘Category: Package orthopoly’ ‘Category: Gamma and factorial functions’
The spherical Bessel function of the first kind.
Reference: Abramowitz and Stegun, equations 10.1.8, page 437 and 10.1.15, page 439.
‘Category: Package orthopoly’ ‘Category: Bessel functions’
The spherical Bessel function of the second kind.
Reference: Abramowitz and Stegun, equations 10.1.9, page 437 and 10.1.15, page 439.
‘Category: Package orthopoly’ ‘Category: Bessel functions’
The spherical Hankel function of the first kind.
Reference: Abramowitz and Stegun, equation 10.1.36, page 439.
‘Category: Package orthopoly’ ‘Category: Bessel functions’
The spherical Hankel function of the second kind.
Reference: Abramowitz and Stegun, equation 10.1.17, page 439.
‘Category: Package orthopoly’ ‘Category: Bessel functions’
The spherical harmonic function.
Reference: Merzbacher 9.64.
‘Category: Package orthopoly’
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 hstep
.
‘Category: Package orthopoly’ ‘Category: Mathematical functions’
The ultraspherical polynomial (also known as the Gegenbauer polynomial).
Reference: Abramowitz and Stegun, equation 22.5.46, page 779.
‘Category: Package orthopoly’
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