Mathematical Applications
Contents
- Summation of Rational Functions
- Mellin-Barnes Integrals
-
n
-Dimensional Sphere
Summation of Rational Functions
As shown in
Temme(1996)
(ยง3.4), the results given in
Series Expansions
can be used to sum infinite series of rational functions.
Example
S
=
∑
k
=
0
∞
a
k
,
a
k
=
k
(
3
k
+
2
)
(
2
k
+
1
)
(
k
+
1
)
By decomposition into partial fractions
a
k
=
2
k
+
2
3
-
1
k
+
1
2
-
1
k
+
1
=
(
1
k
+
1
-
1
k
+
1
2
)
-
2
(
1
k
+
1
-
1
k
+
2
3
)
Hence from (
Series Expansions 6
), ( Special Values and Extrema
Equation 13
and
Equation 19
)
S
=
ψ
(
1
2
)
-
2
ψ
(
2
3
)
-
γ
=
3
ln
3
-
2
ln
2
-
1
3
π
3
Mellin-Barnes Integrals
Many special functions
f
(
z
)
can be represented as a Mellin-Barnes integral, that is,
an integral of a product of gamma functions, reciprocals of gamma
functions, and a power of
z
, the integration contour being doubly-infinite and eventually
parallel to the imaginary axis. The left-hand side of (
Integral Equation 1
) is a typical example. By translating the contour parallel to itself
and summing the residues of the integrand, asymptotic expansions of
f
(
z
)
for large
|
z
|
, or small
|
z
|
, can be obtained complete with an integral representation of the
error term.
n
-Dimensional Sphere
The volume
V
and surface area
A
of the
n
-dimensional sphere of radius
r
are given by
V
=
π
1
2
n
r
n
Γ
(
1
2
n
+
1
)
,
S
=
2
π
1
2
n
r
n
-
1
Γ
(
1
2
n
)
=
n
r
V
