Previous: Introduction to orthogonal polynomials, Up: orthopoly [Contents][Index]
The associated Legendre function of the first kind of degree \(n\) and order \(m\), \(P_{n}^{m}(z)\) , is a solution of the differential equation:
This is related to the Legendre polynomial, \(P_n(x)\) via
Reference: A&S eqn 22.5.37, A&S eqn 8.6.6, and A&S eqn 8.2.5.
Some examples:
(%i1) assoc_legendre_p(2,0,x); 2 3 (1 - x) (%o1) (- 3 (1 - x)) + ---------- + 1 2 (%i2) factor(%); 2 3 x - 1 (%o2) -------- 2 (%i3) factor(assoc_legendre_p(2,1,x)); 2 (%o3) - 3 x sqrt(1 - x ) (%i4) (-1)^1*(1-x^2)^(1/2)*diff(legendre_p(2,x),x); 2 (%o4) - (3 - 3 (1 - x)) sqrt(1 - x ) (%i5) factor(%); 2 (%o5) - 3 x sqrt(1 - x )
The associated Legendre function of the second kind of degree \(n\) and order \(m\), \(Q_{n}^{m}(z)\) , is a solution of the differential equation:
Reference: Abramowitz and Stegun, equation 8.5.3 and 8.1.8.
Some examples:
(%i1) assoc_legendre_q(0,0,x); x + 1 log(- -----) x - 1 (%o1) ------------ 2 (%i2) assoc_legendre_q(1,0,x); x + 1 log(- -----) x - 2 x - 1 (%o2)/R/ ------------------ 2 (%i3) assoc_legendre_q(1,1,x); (%o3)/R/ x + 1 2 2 2 x + 1 2 log(- -----) sqrt(1 - x ) x - 2 sqrt(1 - x ) x - log(- -----) sqrt(1 - x ) x - 1 x - 1 - --------------------------------------------------------------------------- 2 2 x - 2
The Chebyshev polynomial of the first kind of degree \(n\), \(T_n(x).\)
Reference: A&S eqn 22.5.47.
The polynomials \(T_n(x)\) can be written in terms of a hypergeometric function:
The polynomials can also be defined in terms of the sum
or the Rodrigues formula
where
Some examples:
(%i1) chebyshev_t(2,x); 2 (%o1) (- 4 (1 - x)) + 2 (1 - x) + 1 (%i2) factor(%); 2 (%o2) 2 x - 1 (%i3) factor(chebyshev_t(3,x)); 2 (%o3) x (4 x - 3) (%i4) factor(hgfred([-3,3],[1/2],(1-x)/2)); 2 (%o4) x (4 x - 3)
The Chebyshev polynomial of the second kind of degree \(n\), \(U_n(x)\) .
Reference: A&S eqn 22.5.48.
The polynomials \(U_n(x)\) can be written in terms of a hypergeometric function:
The polynomials can also be defined in terms of the sum
or the Rodrigues formula
where
.
(%i1) chebyshev_u(2,x); 2 8 (1 - x) 4 (1 - x) (%o1) 3 ((- ---------) + ---------- + 1) 3 3 (%i2) expand(%); 2 (%o2) 4 x - 1 (%i3) expand(chebyshev_u(3,x)); 3 (%o3) 8 x - 4 x (%i4) expand(4*hgfred([-3,5],[3/2],(1-x)/2)); 3 (%o4) 8 x - 4 x
The generalized Laguerre polynomial of degree \(n\), \(L_n^{(\alpha)}(x)\) .
These can be defined by
The polynomials can also be defined by the sum
or the Rodrigues formula
where
Reference: A&S eqn 22.5.54.
Some examples:
(%i1) gen_laguerre(1,k,x); x (%o1) (k + 1) (1 - -----) k + 1 (%i2) gen_laguerre(2,k,x); 2 x 2 x (k + 1) (k + 2) (--------------- - ----- + 1) (k + 1) (k + 2) k + 1 (%o2) --------------------------------------------- 2 (%i3) binomial(2+k,2)*hgfred([-2],[1+k],x); 2 x 2 x (k + 1) (k + 2) (--------------- - ----- + 1) (k + 1) (k + 2) k + 1 (%o3) --------------------------------------------- 2
The Hermite polynomial of degree \(n\), \(H_n(x)\) .
These polynomials may be defined by a hypergeometric function
or by the series
or the Rodrigues formula
where
Reference: A&S eqn 22.5.55.
Some examples:
(%i1) hermite(3,x); 2 2 x (%o1) - 12 x (1 - ----) 3 (%i2) expand(%); 3 (%o2) 8 x - 12 x (%i3) expand(hermite(4,x)); 4 2 (%o3) 16 x - 48 x + 12 (%i4) expand((2*x)^4*hgfred([-2,-2+1/2],[],-1/x^2)); 4 2 (%o4) 16 x - 48 x + 12 (%i5) expand(4!*sum((-1)^k*(2*x)^(4-2*k)/(k!*(4-2*k)!),k,0,floor(4/2))); 4 2 (%o5) 16 x - 48 x + 12
Return true
if the input is an interval and return false if it isn’t.
The Jacobi polynomial, \(P_n^{(a,b)}(x)\) .
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 \(a \le -1\) or \(b \le -1\) .
Reference: A&S eqn 22.5.42.
The polynomial may be defined in terms of hypergeometric functions:
or the Rodrigues formula
where
Some examples:
(%i1) jacobi_p(0,a,b,x); (%o1) 1 (%i2) jacobi_p(1,a,b,x); (b + a + 2) (1 - x) (%o2) (a + 1) (1 - -------------------) 2 (a + 1)
The Laguerre polynomial, \(L_n(x)\) of degree \(n\).
Reference: A&S eqn 22.5.16 and A&S eqn 22.5.54.
These are related to the generalized Laguerre polynomial by
The polynomials are given by the sum
Some examples:
(%i1) laguerre(1,x); (%o1) 1 - x (%i2) laguerre(2,x); 2 x (%o2) -- - 2 x + 1 2 (%i3) gen_laguerre(2,0,x); 2 x (%o3) -- - 2 x + 1 2 (%i4) sum((-1)^k/k!*binomial(2,k)*x^k,k,0,2); 2 x (%o4) -- - 2 x + 1 2
The Legendre polynomial of the first kind, \(P_n(x)\) , of degree \(n\).
Reference: A&S eqn 22.5.50 and A&S eqn 22.5.51.
The Legendre polynomial is related to the Jacobi polynomials by
or the Rodrigues formula
where
Some examples:
(%i1) legendre_p(1,x); (%o1) x (%i2) legendre_p(2,x); 2 3 (1 - x) (%o2) (- 3 (1 - x)) + ---------- + 1 2 (%i3) expand(%); 2 3 x 1 (%o3) ---- - - 2 2 (%i4) expand(legendre_p(3,x)); 3 5 x 3 x (%o4) ---- - --- 2 2 (%i5) expand(jacobi_p(3,0,0,x)); 3 5 x 3 x (%o5) ---- - --- 2 2
The Legendre function of the second kind, \(Q_n(x)\) of degree \(n\).
Reference: Abramowitz and Stegun, equations 8.5.3 and 8.1.8.
These are related to \(Q_n^m(x)\) by
Some examples:
(%i1) legendre_q(0,x); x + 1 log(- -----) x - 1 (%o1) ------------ 2 (%i2) legendre_q(1,x); x + 1 log(- -----) x - 2 x - 1 (%o2)/R/ ------------------ 2 (%i3) assoc_legendre_q(1,0,x); x + 1 log(- -----) x - 2 x - 1 (%o3)/R/ ------------------ 2
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);
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.
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.
The Pochhammer symbol, \((x)_n\) . (See A&S eqn 6.1.22 and DLMF 5.2.iii).
For nonnegative
integers n with n <= pochhammer_max_index
, the
expression
\((x)_n\)
evaluates to the
product
\(x(x+1)(x+2)\cdots(x+n-1)\)
when
\(n > 0\)
and
to 1 when \(n = 0\).
For negative \(n\),
\((x)_n\)
is
defined as
\((-1)^n/(1-x)_{-n}.\)
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 A&S eqn 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
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: A&S eqn 6.1.16.
The spherical Bessel function of the first kind, \(j_n(x).\)
Reference: A&S eqn 10.1.8 and A&S eqn 10.1.15.
It is related to the Bessel function by
Some examples:
(%i1) spherical_bessel_j(1,x); sin(x) ------ - cos(x) x (%o1) --------------- x (%i2) spherical_bessel_j(2,x); 3 3 cos(x) (- (1 - --) sin(x)) - -------- 2 x x (%o2) ------------------------------ x (%i3) expand(%); sin(x) 3 sin(x) 3 cos(x) (%o3) (- ------) + -------- - -------- x 3 2 x x (%i4) expand(sqrt(%pi/(2*x))*bessel_j(2+1/2,x)),besselexpand:true; sin(x) 3 sin(x) 3 cos(x) (%o4) (- ------) + -------- - -------- x 3 2 x x
The spherical Bessel function of the second kind, \(y_n(x).\)
Reference: A&S eqn 10.1.9 and A&S eqn 10.1.15.
It is related to the Bessel function by
(%i1) spherical_bessel_y(1,x); cos(x) (- sin(x)) - ------ x (%o1) ------------------- x (%i2) spherical_bessel_y(2,x); 3 sin(x) 3 -------- - (1 - --) cos(x) x 2 x (%o2) - -------------------------- x (%i3) expand(%); 3 sin(x) cos(x) 3 cos(x) (%o3) (- --------) + ------ - -------- 2 x 3 x x (%i4) expand(sqrt(%pi/(2*x))*bessel_y(2+1/2,x)),besselexpand:true; 3 sin(x) cos(x) 3 cos(x) (%o4) (- --------) + ------ - -------- 2 x 3 x x
The spherical Hankel function of the first kind, \(h_n^{(1)}(x).\)
Reference: A&S eqn 10.1.36.
This is defined by
The spherical Hankel function of the second kind, \(h_n^{(2)}(x).\)
Reference: A&S eqn 10.1.17.
This is defined by
The spherical harmonic function, \(Y_n^m(\theta, \phi)\) .
Spherical harmonics satisfy the angular part of Laplace’s equation in spherical coordinates.
For integers \(n\) and \(m\) such that \(n \geq |m|\) and for \(\theta \in [0, \pi]\) , Maxima’s spherical harmonic function can be defined by
Further, when \(n < |m|\) , the spherical harmonic function vanishes.
The factor \((-1)^m\), frequently used in Quantum mechanics, is called the Condon-Shortely phase. Some references, including NIST Digital Library of Mathematical Functions omit this factor; see http://dlmf.nist.gov/14.30.E1.
Reference: Merzbacher 9.64.
Some examples:
(%i1) spherical_harmonic(1,0,theta,phi); sqrt(3) cos(theta) (%o1) ------------------ 2 sqrt(%pi) (%i2) spherical_harmonic(1,1,theta,phi); %i phi sqrt(3) %e sin(theta) (%o2) --------------------------- 3/2 2 sqrt(%pi) (%i3) spherical_harmonic(1,-1,theta,phi); - %i phi sqrt(3) %e sin(theta) (%o3) - ----------------------------- 3/2 2 sqrt(%pi) (%i4) spherical_harmonic(2,0,theta,phi); 2 3 (1 - cos(theta)) sqrt(5) ((- 3 (1 - cos(theta))) + ------------------- + 1) 2 (%o4) ---------------------------------------------------------- 2 sqrt(%pi) (%i5) factor(%); 2 sqrt(5) (3 cos (theta) - 1) (%o5) --------------------------- 4 sqrt(%pi)
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
.
The ultraspherical polynomial, \(C_n^{(a)}(x)\) (also known as the Gegenbauer polynomial).
Reference: A&S eqn 22.5.46.
These polynomials can be given in terms of Jacobi polynomials:
or the series
or the Rodrigues formula
where
Some examples:
(%i1) ultraspherical(1,a,x); (2 a + 1) (1 - x) (%o1) 2 a (1 - -----------------) 1 2 (a + -) 2 (%i2) factor(%); (%o2) 2 a x (%i3) factor(ultraspherical(2,a,x)); 2 2 (%o3) a (2 a x + 2 x - 1)
Previous: Introduction to orthogonal polynomials, Up: orthopoly [Contents][Index]