3.4 Associate Legendre Functions with Complex Indices
53
-2
5.3
-1
5.2
20
0
10
9
5.1
15
Graph: P -0.5+100i
m
(x)
m
1
x
5
10
2
4.9
5
4.8
0
Fig. 3.4 Graph of the Mehler or conical functions P m
−0.5+100i (x). The function values were
calculated via erg = mehlerfun(100,m,x).value. The figure is based on 18,750 function
values, which takes less than 3.7 s on a rather old personal computer. The graph uncovers a strong
oscillatory behavior
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−1/2) 2 +τ 2
(n+μ)n
and for second
one ta(n) (2) =
(n−1/2−μ) 2 +τ 2
(n−μ)n
.
Equation (3.23e); method 7: The radius of convergence is |z| > 1. Equation (3.23e) is undefined for τ = 0. Again, we have two summands, Eq. (3.26),
ta(n)
(1)
=
(1/4 − μ/2 + iτ/2 + n − 1) · (3/4 − μ/2 + iτ/2 + n − 1)
n(n + iτ )
, and
ta(n)
(2)
= ta(n)
(1)∗ ,
and for z real, the corresponding prefactors are as well complex conjugated to each
other, and thus the Mehler function P
μ
−
1
2 +iτ
(z) remains for real arguments real
valued.
In case the hypergeometric series expansion should not sufficiently well converge, the recurrence relation (3.29) will be used or the path integration method as
described in Chap. 8 will be used based on the hypergeometric series expansion with
the smallest absolute value of its argument.
53
-2
5.3
-1
5.2
20
0
10
9
5.1
15
Graph: P -0.5+100i
m
(x)
m
1
x
5
10
2
4.9
5
4.8
0
Fig. 3.4 Graph of the Mehler or conical functions P m
−0.5+100i (x). The function values were
calculated via erg = mehlerfun(100,m,x).value. The figure is based on 18,750 function
values, which takes less than 3.7 s on a rather old personal computer. The graph uncovers a strong
oscillatory behavior
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−1/2) 2 +τ 2
(n+μ)n
and for second
one ta(n) (2) =
(n−1/2−μ) 2 +τ 2
(n−μ)n
.
Equation (3.23e); method 7: The radius of convergence is |z| > 1. Equation (3.23e) is undefined for τ = 0. Again, we have two summands, Eq. (3.26),
ta(n)
(1)
=
(1/4 − μ/2 + iτ/2 + n − 1) · (3/4 − μ/2 + iτ/2 + n − 1)
n(n + iτ )
, and
ta(n)
(2)
= ta(n)
(1)∗ ,
and for z real, the corresponding prefactors are as well complex conjugated to each
other, and thus the Mehler function P
μ
−
1
2 +iτ
(z) remains for real arguments real
valued.
In case the hypergeometric series expansion should not sufficiently well converge, the recurrence relation (3.29) will be used or the path integration method as
described in Chap. 8 will be used based on the hypergeometric series expansion with
the smallest absolute value of its argument.