54
3 Legendre Polynomials and Legendre Functions
Conical or Mehler Function of Second Kind
The evaluation of the Mehler function of second kind is also based on hyperspherical series expansions and the path integration method of the hypergeometric
differential equation.
Equation (3.24a); method 1: This series converges for |z| > 1 with ta(n) =
(1/4+μ/2+iτ/2+n−1)·(3/4+μ/2+iτ/2+n−1)
n(n+iτ )
, Eq. (3.26).
Equation (3.24b); method 2: Here, we have a combination of two hypergeometric
series with the same nominator, ta(n) ± =
1
4
(2n−1) 2 +4τ 2 )
n(n±μ)
, and the denominator
differs by ±μ. The series converges for |1 − z| < 2 and is undefined for μ positive
or negative integer. This case will be covered by methods 1 and 3.
Equation (3.24c); method 3: For the two series of method 3, we get
ta(n)
(1)
=
1
16
(4n − 3 − 2μ) 2 + τ 2
n(n − 0.5)
, and
ta(n)
(2)
=
1
16
(4n − 1 − 2μ) 2 + τ 2
n(n + 0.5)
,
and |z| < 1.
Equation (3.24d); for −1 < x < +1: This expansion is restricted to x-values
with
ta(n)
(1)
=
1
16
(4n − 3 + 2μ) 2 + τ 2
n(n − 0.5)
, and
ta(n)
(2)
=
1
16
(4n − 1 + 2μ) 2 + τ 2
n(n + 0.5)
.
If possible the special values, as listed above, will be used to calculate
Q
μ
−
1
2 +iτ
(z). In case none of the hypergeometric expansions converge, Eq. (3.12b)
will be tested for μ non-integer or the path integration method of the hypergeometric
differential equation will be used.
Toroidal or Ring Functions
Toroidal or ring functions are, e.g., used in the solution of the Dirichlet problem with
boundary conditions on a torus, in applications in plasma physics (magnetic fields
in stellarators or tokamaks), astrophysics, or gravitational physics [2]. The toroidal
functions are the associate Legendre functions with ν = −
1
2 + τ , τ ∈ N and μ real,
and z > 1 real
P
μ
−
1
2 +τ
(z)
and
Q
μ
−
1
2 +τ
(z),
and can be evaluated with the class toroidalfun,
>> [obj, res] = toroidalfun(tau,mu,z,PorQ).
3 Legendre Polynomials and Legendre Functions
Conical or Mehler Function of Second Kind
The evaluation of the Mehler function of second kind is also based on hyperspherical series expansions and the path integration method of the hypergeometric
differential equation.
Equation (3.24a); method 1: This series converges for |z| > 1 with ta(n) =
(1/4+μ/2+iτ/2+n−1)·(3/4+μ/2+iτ/2+n−1)
n(n+iτ )
, Eq. (3.26).
Equation (3.24b); method 2: Here, we have a combination of two hypergeometric
series with the same nominator, ta(n) ± =
1
4
(2n−1) 2 +4τ 2 )
n(n±μ)
, and the denominator
differs by ±μ. The series converges for |1 − z| < 2 and is undefined for μ positive
or negative integer. This case will be covered by methods 1 and 3.
Equation (3.24c); method 3: For the two series of method 3, we get
ta(n)
(1)
=
1
16
(4n − 3 − 2μ) 2 + τ 2
n(n − 0.5)
, and
ta(n)
(2)
=
1
16
(4n − 1 − 2μ) 2 + τ 2
n(n + 0.5)
,
and |z| < 1.
Equation (3.24d); for −1 < x < +1: This expansion is restricted to x-values
with
ta(n)
(1)
=
1
16
(4n − 3 + 2μ) 2 + τ 2
n(n − 0.5)
, and
ta(n)
(2)
=
1
16
(4n − 1 + 2μ) 2 + τ 2
n(n + 0.5)
.
If possible the special values, as listed above, will be used to calculate
Q
μ
−
1
2 +iτ
(z). In case none of the hypergeometric expansions converge, Eq. (3.12b)
will be tested for μ non-integer or the path integration method of the hypergeometric
differential equation will be used.
Toroidal or Ring Functions
Toroidal or ring functions are, e.g., used in the solution of the Dirichlet problem with
boundary conditions on a torus, in applications in plasma physics (magnetic fields
in stellarators or tokamaks), astrophysics, or gravitational physics [2]. The toroidal
functions are the associate Legendre functions with ν = −
1
2 + τ , τ ∈ N and μ real,
and z > 1 real
P
μ
−
1
2 +τ
(z)
and
Q
μ
−
1
2 +τ
(z),
and can be evaluated with the class toroidalfun,
>> [obj, res] = toroidalfun(tau,mu,z,PorQ).