3.4 Associate Legendre Functions with Complex Indices
57
and thus ta(n) =
(−0.5+μ+n)(−0.5+τ +μ+n)
n(n+τ )
. This series converges for z > 1 and will
serve as the basis for the path integration ansatz.
3.4.3 Complex Legendre Functions
In this last subsection, we will derive code to compute Legendre functions of first
and second kind with arbitrary arguments and parameters.
The associate Legendre functions are solutions of the differential equation (3.10) with complex indices “ν” (degree) and “μ” (order) and complex
arguments “z.” Of interest are the solutions for −1 < z < +1 and
complex z-values. The latter are multivalued. For z ∈ C P
μ
ν (z) and Q
μ
ν (z)
are those solutions of equation (3.10), which are one-valued and regular
for (z) > 1. The values of the branch cut from z = −1 · · · + 1
are
P
μ
ν (x ± i · 0)
and
Q
μ
ν (x ± i · 0) .
(3.36)
If z is a real number, −1 < z < +1, the two linearly independent solutions P
μ
ν (z)
(first kind) and Q
μ
ν (z) (second kind) transform to
P
μ
ν (z) =
1
2
exp(
1
2
μπi)P
μ
ν (x + i0) + exp(−
1
2
μπi)P
μ
ν (x − i0)
and
(3.37)
Q
μ
ν (z) =
1
2
exp(−μπi)
exp(−
1
2
μπi)Q
μ
ν (x + i0) + exp(
1
2
μπi)Q
μ
ν (x − i0)
,
(3.38)
where x ± i0 denotes the values on the cut. With the following simple example, we
gain a better understanding of this limiting process. legQ3 and legQr are private
functions 1 based on equations (3.24c) and (3.24d).
z = rand; % z and x are the position at which Q_nu^mu
x = z;
% will be computed
mu = rand;
nu = -0.5;
zi = 0.01 * ones(1,10);
zi = cumprod(zi);
z =[z-i * zi , z+i * fliplr(zi)];
% z -> x +- i0
% legQ3 and legQr are two private functions (from class
% PQnumufun), used to uncover the limes process
Res = legQ3(nu,mu,z);
% solution in the z-domain
plot(imag(z),real(Res)),shg
% visualization
1 Private functions are functions that live in the subdirectory “private.” They can only be called
from the parent directory.