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3 Legendre Polynomials and Legendre Functions
Prolate Spheroidal Functions
The functions P m
n (x) and Q m
n (x) with x ∈ R and x > 1 are called prolate spheroidal
functions, which can be easily calculated using the class “PQnumufun.”
Oblate Spheroidal Functions P m
n (ix) and Q m
n (ix) with x ∈ R and x > 0 are
called oblate spheroidal functions. Their function values can be real or complex. To
obtain real-valued functions, a set of new functions is defined via
R
m
n (x) = exp(−i
πn
2
)P
m
n (ix) , and
(3.40)
T
m
n (x) = i exp(i
πn
2
)Q
m
n (ix).
(3.41)
R and T can be computed with either the function [resr, res, RorTinfo,
obj] = RTnm(nin,min,x,RorT) or the PQnumufun method RTnm. For the
function, the different variables are “nin” integer degree of the oblate spheroidal
function (associate Legendre function), “min” integer order of the oblate spheroidal
function (associate Legendre function) “x” real input value with x > 0. Please note
that the argument is purely imaginary for the oblate spheroidal functions, and thus
“x” will be internally multiplied with the imaginary unit i. “RorT” could be 1 or “R”
for oblate spheroidal function of first kind (default), or 2 or “T” for oblate spheroidal
function of second kind. The output are “resr,” the real numerical result, and “res,”
the true numerical result, which could have a small imaginary contribution due to the
finite computational accuracy. “RorTinfo” the information if the oblate spheroidal
function of first or second kind was computed and “obj” the PQnumufun object used
for the computation.
Example
nin = 5; min = 2; x = linspace(0,2,25);
[resr, res, RorTinfo, obj] = RTnm(nin,min,x);
The same example as method
[resr, obj] = PQnumufun(nin,min,i * x).RTnm
This allows directly to use all methods of PQnumufun, e.g., plot(obj,
‘real’, ‘yes’). The object “obj” carries in this case the information
“oblate spheroidal function . . . ” and the function values of the oblate spheroidal
function R or T .