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3 Legendre Polynomials and Legendre Functions
• The SPECFUNPHYS class legendrepoly returns the polynomial coefficients of the Legendre polynomials based on Eq. (3.5) and comes with
methods, e.g., to plot the polynomials and compute the corresponding values.
• legendre2poly returns the polynomial coefficients of the Legendre functions Q n of second kind and has some additional methods for computing and
visualizing Q n .
• The SPECFUNPHYS class Pl0 allows to compute directly the values of the
Legendre polynomial of degree n of first kind and second kind based on
Eq. (3.5).
• The SPECFUNPHYS function JacobiPl returns the nodes of the Legendre
polynomials of first kind based on a Jacobi matrix ansatz.
Associate Legendre function of integer indices:
• The SPECFUNPHYS function JacobiPlm returns the nodes of the associate
Legendre functions of first kind based on a Jacobi matrix ansatz.
• The SPECFUNPHYS class legendrefun returns the polynomial structure of
the associate Legendre function of first kind and allows via additional methods
the computation of functions values and supports plotting.
• The SPECFUNPHYS class Plm returns directly the values of the associate
Legendre function.
Spherical harmonics: The SPECFUNPHYS function Ylm = shfun(l,m,theta,phi) returns the values of the spherical harmonic Y
m
l (θ, φ), and
plotsh(l,m,ax) creates a surface plot of Y
m
l .
Functions with complex indices:
• The SPECFUNPHYS class [obj, resa] = mehlerfun(tauin,muin,z,PorQ) returns the function values of the Mehler or conical function
of first and second kind and supports additional methods like line and surface
plots via the superclass PQnumu, see PQnumufun.
• The SPECFUNPHYS class [obj, resa] = toroidalfun(tauin,
muin,z,PorQ) allows the computation of the toroidal or ring function.
Additional methods are supported via the superclass PQnumu, see
PQnumufun.
• Legendre functions of first and second kind with arbitrary complex indices
and arguments can be computed with the SPECFUNPHYS class [obj,
resa] = PQnumufun(nuin,muin,z,PorQ). PQnumufun supports
the following additional methods (via superclass PQnumu): computing
the absolute function value of an object (abs(obj)), the real part of
the function value (real(obj)), the phase angle of the function value
(angle(obj,dr)), answering the logical question isreal(obj), and
isrealsingle(obj,tol) for taking care of the finite accuracy of the
computation tolerance, and it supports line plots with the overloaded method
plot and surface plots with the overloaded method surf.
3 Legendre Polynomials and Legendre Functions
• The SPECFUNPHYS class legendrepoly returns the polynomial coefficients of the Legendre polynomials based on Eq. (3.5) and comes with
methods, e.g., to plot the polynomials and compute the corresponding values.
• legendre2poly returns the polynomial coefficients of the Legendre functions Q n of second kind and has some additional methods for computing and
visualizing Q n .
• The SPECFUNPHYS class Pl0 allows to compute directly the values of the
Legendre polynomial of degree n of first kind and second kind based on
Eq. (3.5).
• The SPECFUNPHYS function JacobiPl returns the nodes of the Legendre
polynomials of first kind based on a Jacobi matrix ansatz.
Associate Legendre function of integer indices:
• The SPECFUNPHYS function JacobiPlm returns the nodes of the associate
Legendre functions of first kind based on a Jacobi matrix ansatz.
• The SPECFUNPHYS class legendrefun returns the polynomial structure of
the associate Legendre function of first kind and allows via additional methods
the computation of functions values and supports plotting.
• The SPECFUNPHYS class Plm returns directly the values of the associate
Legendre function.
Spherical harmonics: The SPECFUNPHYS function Ylm = shfun(l,m,theta,phi) returns the values of the spherical harmonic Y
m
l (θ, φ), and
plotsh(l,m,ax) creates a surface plot of Y
m
l .
Functions with complex indices:
• The SPECFUNPHYS class [obj, resa] = mehlerfun(tauin,muin,z,PorQ) returns the function values of the Mehler or conical function
of first and second kind and supports additional methods like line and surface
plots via the superclass PQnumu, see PQnumufun.
• The SPECFUNPHYS class [obj, resa] = toroidalfun(tauin,
muin,z,PorQ) allows the computation of the toroidal or ring function.
Additional methods are supported via the superclass PQnumu, see
PQnumufun.
• Legendre functions of first and second kind with arbitrary complex indices
and arguments can be computed with the SPECFUNPHYS class [obj,
resa] = PQnumufun(nuin,muin,z,PorQ). PQnumufun supports
the following additional methods (via superclass PQnumu): computing
the absolute function value of an object (abs(obj)), the real part of
the function value (real(obj)), the phase angle of the function value
(angle(obj,dr)), answering the logical question isreal(obj), and
isrealsingle(obj,tol) for taking care of the finite accuracy of the
computation tolerance, and it supports line plots with the overloaded method
plot and surface plots with the overloaded method surf.