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18
Orthogonal Polynomials
Applications
18.39
Applications in the Physical Sciences
18.39
Applications in the Physical Sciences
18.40
Methods of Computation
Figure 18.39.2
(See
in context
.)
Figure 18.39.2:
Coulomb–Pollaczek weight functions,
x
∈
[
−
1
,
1
]
, (
18.39.50
) for
s
=
10
,
l
=
0
, and
Z
=
±
1
. For
Z
=
+
1
the weight function, red curve, has an essential singularity at
x
=
−
1
, as all derivatives vanish as
x
→
−
1
+
; the green curve is
∫
−
1
x
w
CP
(
y
)
d
y
, to be compared with its
histogram approximation
in §
18.40(ii)
. For
Z
=
−
1
the weight function, blue curve, is non-zero at
x
=
−
1
, but this point is also an essential singularity as the discrete parts of the weight function of (
18.39.51
) accumulate as
k
→
∞
,
x
k
→
−
1
−
.
ⓘ
Annotations:
Symbols:
[
a
,
b
]
: closed interval
,
d
x
: differential of
x
,
∈
: element of
,
∫
: integral
,
y
: real variable
,
w
(
x
)
: weight function
,
l
: nonnegative integer
,
k
: nonnegative integer
and
x
: real variable
Permalink:
http://dlmf.nist.gov/18.39.F2.mag
Encodings:
Magnified png
,
pdf
See also:
Annotations for
§18.39
and
Ch.18