The Barnesian function
G
{\ displaystyle G}
, typically denoted by, is a function that represents an extension of the super faculties to the complex numbers . It is related to the gamma function , the function and the Glaisher-Kinkelin constants and is named after the mathematician Ernest William Barnes .
G
(
z
)
{\ displaystyle G (z)}
K
{\ displaystyle K}
Formally, the Barnesian function is defined in the form of a Weierstrass product as
G
{\ displaystyle G}
G
(
z
+
1
)
=
(
2
π
)
z
/
2
e
-
[
z
(
z
+
1
)
+
γ
z
2
]
/
2
∏
n
=
1
∞
[
(
1
+
z
n
)
n
e
-
z
+
z
2
/
(
2
n
)
]
{\ displaystyle G (z + 1) = (2 \ pi) ^ {z / 2} e ^ {- [z (z + 1) + \ gamma z ^ {2}] / 2} \ prod _ {n = 1} ^ {\ infty} \ left [\ left (1 + {\ frac {z} {n}} \ right) ^ {n} e ^ {- z + z ^ {2} / (2n)} \ right ]}
where denotes the Euler-Mascheroni constant .
γ
{\ displaystyle \ gamma}
Difference equation, functional equation and special values
The Barnesian function satisfies the difference equation
G
{\ displaystyle G}
G
(
z
+
1
)
=
Γ
(
z
)
G
(
z
)
{\ displaystyle \! \ G (z + 1) = \ Gamma (z) G (z)}
with normalization The difference equation implies that takes the following values for integer arguments:
G
(
1
)
=
1.
{\ displaystyle G (1) = 1.}
G
{\ displaystyle G}
G
(
n
)
=
{
0
,
if
n
=
0
,
-
1
,
-
2
,
...
,
∏
i
=
0
n
-
2
i
!
,
if
n
=
1
,
2
,
...
,
{\ displaystyle G (n) = {\ begin {cases} 0, & {\ mbox {falls}} n = 0, -1, -2, \ ldots, \\\ prod _ {i = 0} ^ {n -2} i!, & {\ Mbox {falls}} n = 1,2, \ ldots, \ end {cases}}}
so that
G
(
n
)
=
(
Γ
(
n
)
)
n
-
1
K
(
n
)
{\ displaystyle G (n) = {\ frac {(\ Gamma (n)) ^ {n-1}} {K (n)}}}
where denote the gamma function and the K function . The difference equation uniquely defines the function if the convexity condition is set.
Γ
(
n
)
{\ displaystyle \ Gamma (n)}
K
(
n
)
{\ displaystyle K (n)}
G
{\ displaystyle G}
d
3
d
x
3
G
(
x
)
≥
0
{\ displaystyle {\ frac {\ mathrm {d} ^ {3}} {\ mathrm {d} x ^ {3}}} G (x) \ geq 0}
The difference equation of the -function and the functional equation of the gamma function provide the following functional equation for the -function, as originally proven by Hermann Kinkelin :
G
{\ displaystyle G}
G
{\ displaystyle G}
G
(
1
-
z
)
=
G
(
1
+
z
)
1
(
2
π
)
z
exp
∫
0
z
π
t
cot
π
t
d
t
.
{\ displaystyle G (1-z) = G (1 + z) {\ frac {1} {(2 \ pi) ^ {z}}} \ exp \ int \ limits _ {0} ^ {z} \ pi t \ cot \ pi t \, \ mathrm {d} t.}
Multiplication formula
Like the gamma function, the function also fulfills a multiplication formula:
G
{\ displaystyle G}
G
(
n
z
)
=
K
(
n
)
n
n
2
z
2
/
2
-
n
z
(
2
π
)
-
n
2
-
n
2
z
∏
i
=
0
n
-
1
∏
j
=
0
n
-
1
G
(
z
+
i
+
j
n
)
{\ displaystyle G (nz) = K (n) n ^ {n ^ {2} z ^ {2} / 2-nz} (2 \ pi) ^ {- {\ frac {n ^ {2} -n} {2}} z} \ prod _ {i = 0} ^ {n-1} \ prod _ {j = 0} ^ {n-1} G \ left (z + {\ frac {i + j} {n} } \ right)}
where is a function performed by
K
(
n
)
{\ displaystyle K (n)}
K
(
n
)
=
e
-
(
n
2
-
1
)
ζ
′
(
-
1
)
⋅
n
5
12
⋅
(
2
π
)
(
n
-
1
)
/
2
=
(
A.
e
-
1
12
)
n
2
-
1
⋅
n
5
12
⋅
(
2
π
)
(
n
-
1
)
/
2
.
{\ displaystyle K (n) = e ^ {- (n ^ {2} -1) \ zeta ^ {\ prime} (- 1)} \ cdot n ^ {\ frac {5} {12}} \ cdot ( 2 \ pi) ^ {(n-1) / 2} \, = \, (Ae ^ {- {\ frac {1} {12}}}) ^ {n ^ {2} -1} \ cdot n ^ {\ frac {5} {12}} \ cdot (2 \ pi) ^ {(n-1) / 2}.}
given is. Here is the derivative of the Riemann zeta function and the constant from Glaisher-Kinkelin .
ζ
′
{\ displaystyle \ zeta ^ {\ prime}}
A.
{\ displaystyle A}
Asymptotic development
The function has the following asymptotic expansion found by Barnes:
log
G
(
z
+
1
)
{\ displaystyle \ log \, G (z + 1)}
log
G
(
z
+
1
)
=
1
12
-
log
A.
+
z
2
log
2
π
+
(
z
2
2
-
1
12
)
log
z
-
3
z
2
4th
+
∑
k
=
1
N
B.
2
k
+
2
4th
k
(
k
+
1
)
z
2
k
+
O
(
1
z
2
N
+
2
)
.
{\ displaystyle \ log G (z + 1) = {\ frac {1} {12}} - \ log A + {\ frac {z} {2}} \ log 2 \ pi + \ left ({\ frac {z ^ {2}} {2}} - {\ frac {1} {12}} \ right) \ log z - {\ frac {3z ^ {2}} {4}} + \ sum _ {k = 1} ^ {N} {\ frac {B_ {2k + 2}} {4k \ left (k + 1 \ right) z ^ {2k}}} + O \ left ({\ frac {1} {z ^ {2N + 2}}} \ right).}
Here denotes the Bernoulli numbers and the Glaisher-Kinkelin constant . (Note that at Barnes' time the Bernoulli number was written as . This convention is no longer used.) The expansion is valid for in any sector that does not contain the negative real axis.
B.
k
{\ displaystyle B_ {k}}
A.
{\ displaystyle A}
B.
2
k
{\ displaystyle B_ {2k}}
(
-
1
)
k
+
1
B.
k
{\ displaystyle (-1) ^ {k + 1} B_ {k}}
z
{\ displaystyle z}
Web link
Individual evidence
^ Ernest W. Barnes: The theory of the function
G
{\ displaystyle G}
. In: The Quarterly Journal of Pure and Applied Mathematics , Vol. 31 (1900), pages 264-314.
^ Marie-France Vignéras : L'équation fonctionelle de la fonction zêta de Selberg du groupe modulaire
S.
L.
(
2
,
Z
)
{\ displaystyle SL (2, \ mathbb {Z})}
. In: Astérisque , Vol. 61 (1979), pp. 235-249, ISSN 0303-1179 .
^ Moshe Y. Vardi : Determinants of Laplacians and multiple gamma functions. In: SIAM Journal on Mathematical Analysis , Vol. 19 (1988), pages 493-507, ISSN 0036-1410 .
^ Edmund Taylor Whittaker , George N. Watson: A Course of Modern Analysis. 4th ed. Cambridge University Press, Cambridge 1990, ISBN 978-0-521-09189-3 .
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