orthopoly
assoc_legendre_p (n, m, x) — Function
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:
$$(1-z^2){d^2 w\over dz^2} - 2z{dw\over dz} + \left[n(n+1)-{m^2\over 1-z^2}\right] w = 0$$
$$(1-z^2){d^2 w\over dz^2} - 2z{dw\over dz} + \left[n(n+1)-{m^2\over 1-z^2}\right] w = 0$$
This is related to the Legendre polynomial, $P_n(x)$ via
$$P_n^m(x) = (-1)^m\left(1-x^2\right)^{m/2} {d^m\over dx^m} P_n(x)$$
$$P_n^m(x) = (-1)^m\left(1-x^2\right)^{m/2} {d^m\over dx^m} P_n(x)$$
Reference: https://personal.math.ubc.ca/~cbm/aands/page_779.htmA&S eqn 22.5.37, https://personal.math.ubc.ca/~cbm/aands/page_334.htmA&S eqn 8.6.6, and https://personal.math.ubc.ca/~cbm/aands/page_333.htmA&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 )
See also orthopoly_returns_intervals for how numerical results
are returned.
See also: orthopoly_returns_intervals.
assoc_legendre_q (n, m, x) — Function
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:
$$(1-z^2){d^2 w\over dz^2} - 2z{dw\over dz} + \left[n(n+1)-{m^2\over 1-z^2}\right] w = 0$$
$$(1-z^2){d^2 w\over dz^2} - 2z{dw\over dz} + \left[n(n+1)-{m^2\over 1-z^2}\right] w = 0$$
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
chebyshev_t (n, x) — Function
The Chebyshev polynomial of the first kind of degree $n$, $T_n(x).$
Reference: https://personal.math.ubc.ca/~cbm/aands/page_779.htmA&S eqn 22.5.47.
The polynomials $T_n(x)$ can be written in terms of a hypergeometric function:
$$T_n(x) = {{2}}F{1}\left(-n, n; {1\over 2}; {1-x\over 2}\right)$$
$$T_n(x) = {{2}}F{1}\left(-n, n; {1\over 2}; {1-x\over 2}\right)$$
The polynomials can also be defined in terms of the sum
$$T_n(x) = {n\over 2} \sum_{r=0}^{\lfloor {n/2}\rfloor} {(-1)^r\over n-r} {n-r\choose k}(2x)^{n-2r}$$
$$T_n(x) = {n\over 2} \sum_{r=0}^{\lfloor {n/2}\rfloor} {(-1)^r\over n-r} {n-r\choose k}(2x)^{n-2r}$$
or the Rodrigues formula
$$T_n(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)(1-x^2)^n\right)$$
$$T_n(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)(1-x^2)^n\right)$$
where
$$\eqalign{ w(x) &= 1/\sqrt{1-x^2} \cr \kappa_n &= (-2)^n\left(1\over 2\right)_n }$$
$$\eqalign{ w(x) &= 1/\sqrt{1-x^2} \cr \kappa_n &= (-2)^n\left(1\over 2\right)_n }$$
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)
See also orthopoly_returns_intervals for how numerical results
are returned.
See also: orthopoly_returns_intervals.
chebyshev_u (n, x) — Function
The Chebyshev polynomial of the second kind of degree $n$, $U_n(x).$
Reference: https://personal.math.ubc.ca/~cbm/aands/page_779.htmA&S eqn 22.5.48.
The polynomials $U_n(x)$ can be written in terms of a hypergeometric function:
$$U_n(x) = (n+1); {{2}F{1}}\left(-n, n+2; {3\over 2}; {1-x\over 2}\right)$$
$$U_n(x) = (n+1); {{2}F{1}}\left(-n, n+2; {3\over 2}; {1-x\over 2}\right)$$
The polynomials can also be defined in terms of the sum
$$U_n(x) = \sum_{r=0}^{\lfloor n/2 \rfloor} (-1)^r {n-r \choose r} (2x)^{n-2r}$$
$$U_n(x) = \sum_{r=0}^{\lfloor n/2 \rfloor} (-1)^r {n-r \choose r} (2x)^{n-2r}$$
or the Rodrigues formula
$$U_n(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)(1-x^2)^n\right)$$
$$U_n(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)(1-x^2)^n\right)$$
where
$$\eqalign{ w(x) &= \sqrt{1-x^2} \cr \kappa_n &= {(-2)^n\left({3\over 2}\right)_n \over n+1} }$$
$$\eqalign{ w(x) &= \sqrt{1-x^2} \cr \kappa_n &= {(-2)^n\left({3\over 2}\right)_n \over n+1} }$$
.
(%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
See also orthopoly_returns_intervals for how numerical results
are returned.
See also: orthopoly_returns_intervals.
gen_laguerre (n, a, x) — Function
The generalized Laguerre polynomial of degree $n$, $L_n^{(\alpha)}(x).$
These can be defined by
$$L_n^{(\alpha)}(x) = {n+\alpha \choose n}; {_1F_1}(-n; \alpha+1; x)$$
$$L_n^{(\alpha)}(x) = {n+\alpha \choose n}; {_1F_1}(-n; \alpha+1; x)$$
The polynomials can also be defined by the sum
$$L_n^{(\alpha)}(x) = \sum_{k=0}^n {(\alpha + k + 1)_{n-k} \over (n-k)! k!} (-x)^k$$
$$L_n^{(\alpha)}(x) = \sum_{k=0}^n {(\alpha + k + 1)_{n-k} \over (n-k)! k!} (-x)^k$$
or the Rodrigues formula
$$L_n^{(\alpha)}(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)x^n\right)$$
$$L_n^{(\alpha)}(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)x^n\right)$$
where
$$\eqalign{ w(x) &= e^{-x}x^{\alpha} \cr \kappa_n &= n! }$$
$$\eqalign{ w(x) &= e^{-x}x^{\alpha} \cr \kappa_n &= n! }$$
Reference: https://personal.math.ubc.ca/~cbm/aands/page_780.htmA&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
See also orthopoly_returns_intervals for how numerical results
are returned.
See also: orthopoly_returns_intervals.
hermite (n, x) — Function
The Hermite polynomial of degree $n$, $H_n(x).$
These polynomials may be defined by a hypergeometric function
$$H_n(x) = (2x)^n; {_2F_0}\left(-{1\over 2} n, -{1\over 2}n+{1\over 2};;-{1\over x^2}\right)$$
$$H_n(x) = (2x)^n; {_2F_0}\left(-{1\over 2} n, -{1\over 2}n+{1\over 2};;-{1\over x^2}\right)$$
or by the series
$$H_n(x) = n! \sum_{k=0}^{\lfloor n/2 \rfloor} {(-1)^k(2x)^{n-2k} \over k! (n-2k)!}$$
$$H_n(x) = n! \sum_{k=0}^{\lfloor n/2 \rfloor} {(-1)^k(2x)^{n-2k} \over k! (n-2k)!}$$
or the Rodrigues formula
$$H_n(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)\right)$$
$$H_n(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)\right)$$
where
$$\eqalign{ w(x) &= e^{-{x^2/2}} \cr \kappa_n &= (-1)^n }$$
$$\eqalign{ w(x) &= e^{-{x^2/2}} \cr \kappa_n &= (-1)^n }$$
Reference: https://personal.math.ubc.ca/~cbm/aands/page_780.htmA&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
See also orthopoly_returns_intervals for how numerical results
are returned.
See also: orthopoly_returns_intervals.
intervalp (e) — Function
Return true if the input is an interval and return false if it isn’t.
jacobi_p (n, a, b, x) — Function
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: https://personal.math.ubc.ca/~cbm/aands/page_779.htmA&S eqn 22.5.42.
The polynomial may be defined in terms of hypergeometric functions:
$$P_n^{(a,b)}(x) = {n+a\choose n} {_1F_2}\left(-n, n + a + b + 1; a+1; {1-x\over 2}\right)$$
$$P_n^{(a,b)}(x) = {n+a\choose n} {_1F_2}\left(-n, n + a + b + 1; a+1; {1-x\over 2}\right)$$
or the Rodrigues formula
$$P_n^{(a, b)}(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)\left(1-x^2\right)^n\right)$$
$$P_n^{(a, b)}(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)\left(1-x^2\right)^n\right)$$
where
$$\eqalign{ w(x) &= (1-x)^a(1-x)^b \cr \kappa_n &= (-2)^n n! }$$
$$\eqalign{ w(x) &= (1-x)^a(1-x)^b \cr \kappa_n &= (-2)^n n! }$$
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)
See also orthopoly_returns_intervals for how numerical results
are returned.
See also: orthopoly_returns_intervals.
laguerre (n, x) — Function
The Laguerre polynomial, $L_n(x)$ of degree $n$.
Reference: https://personal.math.ubc.ca/~cbm/aands/page_778.htmA&S eqn 22.5.16 and https://personal.math.ubc.ca/~cbm/aands/page_780.htmA&S eqn 22.5.54.
These are related to the generalized Laguerre polynomial by
$$L_n(x) = L_n^{(0)}(x)$$
$$L_n(x) = L_n^{(0)}(x)$$
The polynomials are given by the sum
$$L_n(x) = \sum_{k=0}^{n} {(-1)^k\over k!}{n \choose k} x^k$$
$$L_n(x) = \sum_{k=0}^{n} {(-1)^k\over k!}{n \choose k} x^k$$
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
See also orthopoly_returns_intervals for how numerical results
are returned.
See also: orthopoly_returns_intervals.
legendre_p (n, x) — Function
The Legendre polynomial of the first kind, $P_n(x),$ of degree $n$.
Reference: https://personal.math.ubc.ca/~cbm/aands/page_779.htmA&S eqn 22.5.50 and https://personal.math.ubc.ca/~cbm/aands/page_779.htmA&S eqn 22.5.51.
The Legendre polynomial is related to the Jacobi polynomials by
$$P_n(x) = P_n^{(0,0)}(x)$$
$$P_n(x) = P_n^{(0,0)}(x)$$
or the Rodrigues formula
$$P_n(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)\left(1-x^2\right)^n\right)$$
$$P_n(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)\left(1-x^2\right)^n\right)$$
where
$$\eqalign{ w(x) &= 1 \cr \kappa_n &= (-2)^n n! }$$
$$\eqalign{ w(x) &= 1 \cr \kappa_n &= (-2)^n n! }$$
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
See also orthopoly_returns_intervals for how numerical results
are returned.
See also: orthopoly_returns_intervals.
legendre_q (n, x) — Function
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
$$Q_n(x) = Q_n^0(x)$$
$$Q_n(x) = Q_n^0(x)$$
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
orthopoly_recur (f, args) — Function
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);
orthopoly_returns_intervals — Variable
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.
See also: true.
orthopoly_weight (f, args) — Function
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.
pochhammer (x, n) — Function
The Pochhammer symbol, $(x)_n.$ (See https://personal.math.ubc.ca/~cbm/aands/page_256.htmA&S eqn 6.1.22 and https://dlmf.nist.gov/5.2.iiiDLMF 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 https://personal.math.ubc.ca/~cbm/aands/page_256.htmA&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
See also: pochhammer_max_index, pochhammer.
pochhammer_max_index — Variable
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: https://personal.math.ubc.ca/~cbm/aands/page_256.htmA&S eqn 6.1.16.
spherical_bessel_j (n, x) — Function
The spherical Bessel function of the first kind, $j_n(x).$
Reference: https://personal.math.ubc.ca/~cbm/aands/page_437.htmA&S eqn 10.1.8 and https://personal.math.ubc.ca/~cbm/aands/page_439.htmA&S eqn 10.1.15.
It is related to the Bessel function by
$$j_n(x) = \sqrt{\pi\over 2x} J_{n+1/2}(x)$$
$$j_n(x) = \sqrt{\pi\over 2x} J_{n+1/2}(x)$$
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
spherical_bessel_y (n, x) — Function
The spherical Bessel function of the second kind, $y_n(x).$
Reference: https://personal.math.ubc.ca/~cbm/aands/page_437.htmA&S eqn 10.1.9 and https://personal.math.ubc.ca/~cbm/aands/page_439.htmA&S eqn 10.1.15.
It is related to the Bessel function by
$$y_n(x) = \sqrt{\pi\over 2x} Y_{n+1/2}(x)$$
$$y_n(x) = \sqrt{\pi\over 2x} Y_{n+1/2}(x)$$
(%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
spherical_hankel1 (n, x) — Function
The spherical Hankel function of the first kind, $h_n^{(1)}(x).$
Reference: https://personal.math.ubc.ca/~cbm/aands/page_439.htmA&S eqn 10.1.36.
This is defined by
$$h_n^{(1)}(x) = j_n(x) + iy_n(x)$$
$$h_n^{(1)}(x) = j_n(x) + iy_n(x)$$
See also orthopoly_returns_intervals for how numerical results
are returned.
See also: orthopoly_returns_intervals.
spherical_hankel2 (n, x) — Function
The spherical Hankel function of the second kind, $h_n^{(2)}(x).$
Reference: https://personal.math.ubc.ca/~cbm/aands/page_439.htmA&S eqn 10.1.17.
This is defined by
$$h_n^{(2)}(x) = j_n(x) + iy_n(x)$$
$$h_n^{(2)}(x) = j_n(x) + iy_n(x)$$
See also orthopoly_returns_intervals for how numerical results
are returned.
See also: orthopoly_returns_intervals.
spherical_harmonic (n, m, theta, phi) — Function
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
$$Y_n^m(\theta, \phi) = (-1)^m \sqrt{{2n+1\over 4\pi} {(n-m)!\over (n+m)!}} P_n^m(\cos\theta) e^{im\phi}$$
$$Y_n^m(\theta, \phi) = (-1)^m \sqrt{{2n+1\over 4\pi} {(n-m)!\over (n+m)!}} P_n^m(\cos\theta) e^{im\phi}$$
Further, when $n < |m|,$ the spherical harmonic function vanishes.
The factor $(-1)^m$, frequently used in Quantum mechanics, is called the https://en.wikipedia.org/wiki/Spherical_harmonics#Condon%E2%80%93Shortley_phaseCondon-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)
See also orthopoly_returns_intervals for how numerical results
are returned.
See also: orthopoly_returns_intervals.
ultraspherical (n, a, x) — Function
The ultraspherical polynomial, $C_n^{(a)}(x)$ (also known as the Gegenbauer polynomial).
Reference: https://personal.math.ubc.ca/~cbm/aands/page_779.htmA&S eqn 22.5.46.
These polynomials can be given in terms of Jacobi polynomials:
$$C_n^{(\alpha)}(x) = {\Gamma\left(\alpha + {1\over 2}\right) \over \Gamma(2\alpha)} {\Gamma(n+2\alpha) \over \Gamma\left(n+\alpha + {1\over 2}\right)} P_n^{(\alpha-1/2, \alpha-1/2)}(x)$$
$$C_n^{(\alpha)}(x) = {\Gamma\left(\alpha + {1\over 2}\right) \over \Gamma(2\alpha)} {\Gamma(n+2\alpha) \over \Gamma\left(n+\alpha + {1\over 2}\right)} P_n^{(\alpha-1/2, \alpha-1/2)}(x)$$
or the series
$$C_n^{(\alpha)}(x) = \sum_{k=0}^{\lfloor n/2 \rfloor} {(-1)^k (\alpha)_{n-k} \over k! (n-2k)!}(2x)^{n-2k}$$
$$C_n^{(\alpha)}(x) = \sum_{k=0}^{\lfloor n/2 \rfloor} {(-1)^k (\alpha)_{n-k} \over k! (n-2k)!}(2x)^{n-2k}$$
or the Rodrigues formula
$$C_n^{(\alpha)}(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)\left(1-x^2\right)^n\right)$$
$$C_n^{(\alpha)}(x) = {1\over \kappa_n w(x)} {d^n\over dx^n}\left(w(x)\left(1-x^2\right)^n\right)$$
where
$$\eqalign{ w(x) &= \left(1-x^2\right)^{\alpha-{1\over 2}} \cr \kappa_n &= {(-2)^n\left(\alpha + {1\over 2}\right)_n n!\over (2\alpha)_n} \cr }$$
$$\eqalign{ w(x) &= \left(1-x^2\right)^{\alpha-{1\over 2}} \cr \kappa_n &= {(-2)^n\left(\alpha + {1\over 2}\right)_n n!\over (2\alpha)_n} \cr }$$
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)
See also orthopoly_returns_intervals for how numerical results
are returned.
See also: orthopoly_returns_intervals.
unit_step (x) — Function
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.
See also: hstep.