80.2 Functions and Variables for orthogonal polynomials

Function: assoc_legendre_p (n, m, x) ¶

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:

\[(1-z^2){d^2 w\over dz^2} - 2z{dw\over dz} + \left[n(n+1)-{m^2\over 1-z^2}\right] w = 0 \]

This is related to the Legendre polynomial, \(P_n(x)\) via

\[P_n^m(x) = (-1)^m\left(1-x^2\right)^{m/2} {d^m\over dx^m} P_n(x) \]

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 )

Categories: Package orthopoly ·

Function: assoc_legendre_q (n, m, 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:

\[(1-z^2){d^2 w\over dz^2} - 2z{dw\over dz} + \left[n(n+1)-{m^2\over 1-z^2}\right] w = 0 \]

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

Categories: Package orthopoly ·

Function: chebyshev_t (n, x) ¶

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:

\[T_n(x) = {_{2}}F_{1}\left(-n, n; {1\over 2}; {1-x\over 2}\right) \]

The polynomials can also be defined in terms of the sum

\[T_n(x) = {n\over 2} \sum_{r=0}^{\lfloor {n/2}\rfloor} {(-1)^r\over n-r} {n-r\choose k}(2x)^{n-2r} \]

or the Rodrigues formula

\[T_n(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)(1-x^2)^n\right) \]

where

\[\eqalign{ w(x) &= 1/\sqrt{1-x^2} \cr \kappa_n &= (-2)^n\left(1\over 2\right)_n } \]

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)

Categories: Package orthopoly ·

Function: chebyshev_u (n, x) ¶

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:

\[U_n(x) = (n+1)\; {_{2}F_{1}}\left(-n, n+2; {3\over 2}; {1-x\over 2}\right) \]

The polynomials can also be defined in terms of the sum

\[U_n(x) = \sum_{r=0}^{\lfloor n/2 \rfloor} (-1)^r {n-r \choose r} (2x)^{n-2r} \]

or the Rodrigues formula

\[U_n(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)(1-x^2)^n\right) \]

where

\[\eqalign{ w(x) &= \sqrt{1-x^2} \cr \kappa_n &= {(-2)^n\left({3\over 2}\right)_n \over n+1} } \]

.

(%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

Categories: Package orthopoly ·

Function: gen_laguerre (n, a, x) ¶

The generalized Laguerre polynomial of degree \(n\), \(L_n^{(\alpha)}(x)\) .

These can be defined by

\[L_n^{(\alpha)}(x) = {n+\alpha \choose n}\; {_1F_1}(-n; \alpha+1; x) \]

The polynomials can also be defined by the sum

\[L_n^{(\alpha)}(x) = \sum_{k=0}^n {(\alpha + k + 1)_{n-k} \over (n-k)! k!} (-x)^k \]

or the Rodrigues formula

\[L_n^{(\alpha)}(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)x^n\right) \]

where

\[\eqalign{ w(x) &= e^{-x}x^{\alpha} \cr \kappa_n &= n! } \]

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

Categories: Package orthopoly ·

Function: hermite (n, x) ¶

The Hermite polynomial of degree \(n\), \(H_n(x)\) .

These polynomials may be defined by a hypergeometric function

\[H_n(x) = (2x)^n\; {_2F_0}\left(-{1\over 2} n, -{1\over 2}n+{1\over 2};;-{1\over x^2}\right) \]

or by the series

\[H_n(x) = n! \sum_{k=0}^{\lfloor n/2 \rfloor} {(-1)^k(2x)^{n-2k} \over k! (n-2k)!} \]

or the Rodrigues formula

\[H_n(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)\right) \]

where

\[\eqalign{ w(x) &= e^{-{x^2/2}} \cr \kappa_n &= (-1)^n } \]

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

Categories: Package orthopoly ·

Function: intervalp (e) ¶

Return true if the input is an interval and return false if it isn’t.

Categories: Package orthopoly · Predicate functions ·

Function: jacobi_p (n, a, b, x) ¶

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:

\[P_n^{(a,b)}(x) = {n+a\choose n} {_1F_2}\left(-n, n + a + b + 1; a+1; {1-x\over 2}\right) \]

or the Rodrigues formula

\[P_n^{(a, b)}(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)\left(1-x^2\right)^n\right) \]

where

\[\eqalign{ w(x) &= (1-x)^a(1-x)^b \cr \kappa_n &= (-2)^n n! } \]

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)

Categories: Package orthopoly ·

Function: laguerre (n, x) ¶

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

\[L_n(x) = L_n^{(0)}(x) \]

The polynomials are given by the sum

\[L_n(x) = \sum_{k=0}^{n} {(-1)^k\over k!}{n \choose k} x^k \]

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

Categories: Package orthopoly ·

Function: legendre_p (n, x) ¶

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

\[P_n(x) = P_n^{(0,0)}(x) \]

or the Rodrigues formula

\[P_n(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)\left(1-x^2\right)^n\right) \]

where

\[\eqalign{ w(x) &= 1 \cr \kappa_n &= (-2)^n n! } \]

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

Categories: Package orthopoly ·

Function: legendre_q (n, x) ¶

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

\[Q_n(x) = Q_n^0(x) \]

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

Categories: Package orthopoly ·

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);

Categories: Package orthopoly ·

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.

Categories: Package orthopoly ·

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.

Categories: Package orthopoly ·

Function: pochhammer (x, n) ¶

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

Categories: Package orthopoly · Gamma and factorial functions ·

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: A&S eqn 6.1.16.

Categories: Package orthopoly · Gamma and factorial functions ·

Function: spherical_bessel_j (n, x) ¶

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

\[j_n(x) = \sqrt{\pi\over 2x} J_{n+1/2}(x) \]

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

Categories: Package orthopoly · Bessel functions ·

Function: spherical_bessel_y (n, 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

\[y_n(x) = \sqrt{\pi\over 2x} Y_{n+1/2}(x) \]

(%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

Categories: Package orthopoly · Bessel functions ·

Function: spherical_hankel1 (n, x) ¶

The spherical Hankel function of the first kind, \(h_n^{(1)}(x).\)

Reference: A&S eqn 10.1.36.

This is defined by

\[h_n^{(1)}(x) = j_n(x) + iy_n(x) \]

Categories: Package orthopoly · Bessel functions ·

Function: spherical_hankel2 (n, x) ¶

The spherical Hankel function of the second kind, \(h_n^{(2)}(x).\)

Reference: A&S eqn 10.1.17.

This is defined by

\[h_n^{(2)}(x) = j_n(x) + iy_n(x) \]

Categories: Package orthopoly · Bessel functions ·

Function: spherical_harmonic (n, m, theta, phi) ¶

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

\[Y_n^m(\theta, \phi) = (-1)^m \sqrt{{2n+1\over 4\pi} {(n-m)!\over (n+m)!}} P_n^m(\cos\theta) e^{im\phi} \]

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)

Categories: Package orthopoly ·

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 hstep.

Categories: Package orthopoly · Mathematical functions ·

Function: ultraspherical (n, a, x) ¶

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:

\[C_n^{(\alpha)}(x) = {\Gamma\left(\alpha + {1\over 2}\right) \over \Gamma(2\alpha)} {\Gamma(n+2\alpha) \over \Gamma\left(n+\alpha + {1\over 2}\right)} P_n^{(\alpha-1/2, \alpha-1/2)}(x) \]

or the series

\[C_n^{(\alpha)}(x) = \sum_{k=0}^{\lfloor n/2 \rfloor} {(-1)^k (\alpha)_{n-k} \over k! (n-2k)!}(2x)^{n-2k} \]

or the Rodrigues formula

\[C_n^{(\alpha)}(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)\left(1-x^2\right)^n\right) \]

where

\[\eqalign{ w(x) &= \left(1-x^2\right)^{\alpha-{1\over 2}} \cr \kappa_n &= {(-2)^n\left(\alpha + {1\over 2}\right)_n n!\over (2\alpha)_n} \cr } \]

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)

Categories: Package orthopoly ·