56
3 Legendre Polynomials and Legendre Functions
Equation (3.23a); method 1: For this hypergeometric series, we get, Eq. (3.26),
ta(n) =
1
4
(2n−1) 2 −4τ 2
(n−μ)n . Here, P
μ
−
1
2 +τ
(z) will be undefined for μ = m with m
positive integer. With Eq. (3.12b), we get
P
m
−
1
2 +τ
(z) =
(−1) m
π
Γ (m + 1/2 + τ )Γ (m + 1/2 − τ )P
−m
−
1
2 +τ
(z) ,
(3.33)
and thus it is sufficient to evaluate P
−m
−
1
2 +τ
(z).
Equation (3.23c); method 2: Here, we get for the nominator (
1
2 +n−μ) 2 −τ 2 and
thus a finite series if τ ≥ −
1
2 − μ and μ =
2m+1
2
with m 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.
Of particular interest is the following formula:
P
μ
−
1
2 +τ
(cosh η) = Γ (1 − μ)
−1 2
2μ (1 − exp(−2η))
−μ exp
−(τ +
1
2
)η
× 2 F 1
1
2
− μ,
1
2
+ τ − μ; 1 − 2μ; 1 − exp(−2η)
. (3.34)
This series will converge for |1 − exp(−2 ∗ η)| < 1. The hypergeometric series of
method 1 will converge for |1 − z| < 2. For z < 2.9 |
1−z
2 | < |1 − exp(−2 ∗ η)|,
and therefore the series expansion (3.34) will only be used as basic for the path
integration method.
Toroidal or Ring Function of Second Kind
Computing the toroidal function of second kind is also based on hyperspherical
series expansions and the path integration method of the hypergeometric differential
equation. We will use two different series expansions called “method 2” and
“method 4” in the “info” property.
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 method 4:
Q
μ
−
1
2 +τ
(cosh η) = exp(iμπ)
√
π
Γ (
1
2 + τ + μ)
Γ (1 + τ )
exp(η) − exp(−η)
μ
× exp
−(
1
2
+ τ + μ)
× 2 F 1
1
2
+ μ,
1
2
+ τ + μ; 1 + τ ; exp(−2η)
,
(3.35)
3 Legendre Polynomials and Legendre Functions
Equation (3.23a); method 1: For this hypergeometric series, we get, Eq. (3.26),
ta(n) =
1
4
(2n−1) 2 −4τ 2
(n−μ)n . Here, P
μ
−
1
2 +τ
(z) will be undefined for μ = m with m
positive integer. With Eq. (3.12b), we get
P
m
−
1
2 +τ
(z) =
(−1) m
π
Γ (m + 1/2 + τ )Γ (m + 1/2 − τ )P
−m
−
1
2 +τ
(z) ,
(3.33)
and thus it is sufficient to evaluate P
−m
−
1
2 +τ
(z).
Equation (3.23c); method 2: Here, we get for the nominator (
1
2 +n−μ) 2 −τ 2 and
thus a finite series if τ ≥ −
1
2 − μ and μ =
2m+1
2
with m 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.
Of particular interest is the following formula:
P
μ
−
1
2 +τ
(cosh η) = Γ (1 − μ)
−1 2
2μ (1 − exp(−2η))
−μ exp
−(τ +
1
2
)η
× 2 F 1
1
2
− μ,
1
2
+ τ − μ; 1 − 2μ; 1 − exp(−2η)
. (3.34)
This series will converge for |1 − exp(−2 ∗ η)| < 1. The hypergeometric series of
method 1 will converge for |1 − z| < 2. For z < 2.9 |
1−z
2 | < |1 − exp(−2 ∗ η)|,
and therefore the series expansion (3.34) will only be used as basic for the path
integration method.
Toroidal or Ring Function of Second Kind
Computing the toroidal function of second kind is also based on hyperspherical
series expansions and the path integration method of the hypergeometric differential
equation. We will use two different series expansions called “method 2” and
“method 4” in the “info” property.
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 method 4:
Q
μ
−
1
2 +τ
(cosh η) = exp(iμπ)
√
π
Γ (
1
2 + τ + μ)
Γ (1 + τ )
exp(η) − exp(−η)
μ
× exp
−(
1
2
+ τ + μ)
× 2 F 1
1
2
+ μ,
1
2
+ τ + μ; 1 + τ ; exp(−2η)
,
(3.35)