60
3 Legendre Polynomials and Legendre Functions
“tol” comes together with a relative tolerance value tol. In case
imag(obj.value)./real(obj.value) < tol,
the corresponding element will be treated as real.
• surf(obj, name,value) will create a surface plot. In case the Legendre
function value will be complex, the absolute value will be plotted and its phase
(angle) color coded. The optional name-value pairs are
“what” with value “z” (default) for a surface plot in the complex domain; “mu” or
“nu” for z, mu and, respectively, nu as axis and the Legendre function value; “sp”
for superposition over all nu/mu and, respectively, tau/mu pairs in the complex
z-domain.
“real” with value “yes” or “no,” see plot.
“row” with an integer value: in case there are multiple numu/taumu rows, which
one should be selected.
“weight” with a real column vector of the length equal the number of numu or
taumu pairs. “weight” will only be supported for “what”, with value ‘sp’ or ‘mu’.
Example:
mehlerfun(tau,mu,z,2).surf(’what’,’mu’,’weight’,we)
Legendre Function of First Kind
Similar to the Mehler function, the computations are based on the hypergeometric
series expansions, or on a path integration method. In the object property “info,” the
different hypergeometric series are referred as method 1, 2, 4, or 7.
Method 1 is based on Eqs. (3.23a) and (3.23b): Here, we get, Eq. (3.26), ta(n) =
(n−ν−1)(n+ν)
n(n−μ)
. P
μ
ν (z) will be undefined for μ = m with m positive integer. With
Eq. (3.12b), we get
P
m
ν (z) =
Γ (ν + m + 1)
Γ (ν − m + 1)
P
−m
ν (z) ,
(3.39)
and thus it is sufficient to evaluate P −m
ν (z).
Equation (3.23c); method 2: Here, we get ta(n) =
(n−ν−μ−1)(n+ν−μ)
n(n−μ)
, and thus a
finite series if either ν + μ or μ − ν becomes a positive integer. Because methods 1
and 2 have the same radius of convergence |1 − z| < 2, method 2 will only be used
for the finite case mentioned above.
Equation (3.23d); method 4: This evaluation is based on a combination of two
hypergeometric series. The radius of convergence is |1 + z| < 2, and the equation
is not defined for integer μ-values, which will be covered either by method 1 or by
method 7. For the first summand, we have ta(n) (1) =
(n−ν−1)(n+ν)
n(n+μ)
and ta(n) (2) =
(n−ν−μ−1)(n+ν−μ)
n(n−μ)
.
3 Legendre Polynomials and Legendre Functions
“tol” comes together with a relative tolerance value tol. In case
imag(obj.value)./real(obj.value) < tol,
the corresponding element will be treated as real.
• surf(obj, name,value) will create a surface plot. In case the Legendre
function value will be complex, the absolute value will be plotted and its phase
(angle) color coded. The optional name-value pairs are
“what” with value “z” (default) for a surface plot in the complex domain; “mu” or
“nu” for z, mu and, respectively, nu as axis and the Legendre function value; “sp”
for superposition over all nu/mu and, respectively, tau/mu pairs in the complex
z-domain.
“real” with value “yes” or “no,” see plot.
“row” with an integer value: in case there are multiple numu/taumu rows, which
one should be selected.
“weight” with a real column vector of the length equal the number of numu or
taumu pairs. “weight” will only be supported for “what”, with value ‘sp’ or ‘mu’.
Example:
mehlerfun(tau,mu,z,2).surf(’what’,’mu’,’weight’,we)
Legendre Function of First Kind
Similar to the Mehler function, the computations are based on the hypergeometric
series expansions, or on a path integration method. In the object property “info,” the
different hypergeometric series are referred as method 1, 2, 4, or 7.
Method 1 is based on Eqs. (3.23a) and (3.23b): Here, we get, Eq. (3.26), ta(n) =
(n−ν−1)(n+ν)
n(n−μ)
. P
μ
ν (z) will be undefined for μ = m with m positive integer. With
Eq. (3.12b), we get
P
m
ν (z) =
Γ (ν + m + 1)
Γ (ν − m + 1)
P
−m
ν (z) ,
(3.39)
and thus it is sufficient to evaluate P −m
ν (z).
Equation (3.23c); method 2: Here, we get ta(n) =
(n−ν−μ−1)(n+ν−μ)
n(n−μ)
, and thus a
finite series if either ν + μ or μ − ν becomes a positive integer. Because methods 1
and 2 have the same radius of convergence |1 − z| < 2, method 2 will only be used
for the finite case mentioned above.
Equation (3.23d); method 4: This evaluation is based on a combination of two
hypergeometric series. The radius of convergence is |1 + z| < 2, and the equation
is not defined for integer μ-values, which will be covered either by method 1 or by
method 7. For the first summand, we have ta(n) (1) =
(n−ν−1)(n+ν)
n(n+μ)
and ta(n) (2) =
(n−ν−μ−1)(n+ν−μ)
n(n−μ)
.