Graph der Clausen-Funktion
C
l
2
(
θ
)
{\displaystyle \mathrm {Cl} _{2}(\theta )}
C
l
4
(
θ
)
{\displaystyle \mathrm {Cl} _{4}(\theta )}
In der Mathematik ist die Clausen-Funktion (nach Thomas Clausen ) durch das folgende Integral definiert:
Cl
2
(
θ
)
=
−
∫
0
θ
log
|
2
sin
(
t
/
2
)
|
d
t
.
{\displaystyle \operatorname {Cl} _{2}(\theta )=-\int _{0}^{\theta }\log |2\sin(t/2)|\,\mathrm {d} t.}
Allgemeiner definiert man für komplexe
s
{\displaystyle s}
Re
(
s
)
>
1
{\displaystyle \operatorname {Re} (s)>1}
Cl
s
(
θ
)
=
∑
n
=
1
∞
sin
(
n
θ
)
n
s
=
sin
(
θ
)
+
sin
(
2
θ
)
2
s
+
sin
(
3
θ
)
3
s
+
sin
(
4
θ
)
4
s
+
⋯
{\displaystyle \operatorname {Cl} _{s}(\theta )=\sum _{n=1}^{\infty }{\frac {\sin(n\theta )}{n^{s}}}=\sin(\theta )+{\frac {\sin(2\theta )}{2^{s}}}+{\frac {\sin(3\theta )}{3^{s}}}+{\frac {\sin(4\theta )}{4^{s}}}+\cdots }
Diese Definition kann auf der gesamten komplexen Ebene analytisch fortgesetzt werden.
Beispiele von Glaisher-Clausen-Funktionen im Intervall
[
0
,
2
⋅
π
]
{\displaystyle [0,\,2\cdot \pi ]}
Beispiele von Standard-Clausen-Funktionen im Intervall
[
0
,
2
⋅
π
]
{\displaystyle [0,\,2\cdot \pi ]}
Eine verallgemeinerte Definition der Clausen-Funktionen für lautet:
C
l
z
(
θ
)
=
{
S
z
(
θ
)
=
∑
k
=
1
∞
sin
(
k
⋅
θ
)
k
z
C
z
(
θ
)
=
∑
k
=
1
∞
cos
(
k
⋅
θ
)
k
z
{\displaystyle \operatorname {Cl_{z}} \left(\theta \right)={\begin{cases}\operatorname {S_{z}} \left(\theta \right)=\sum _{k=1}^{\infty }{\frac {\sin \left(k\cdot \theta \right)}{k^{z}}}\\\operatorname {C_{z}} \left(\theta \right)=\sum _{k=1}^{\infty }{\frac {\cos \left(k\cdot \theta \right)}{k^{z}}}\\\end{cases}}}
[ 1]
Clausen-Funktionen der Form
S
z
(
θ
)
=
∑
k
=
1
∞
sin
(
k
⋅
θ
)
k
z
{\displaystyle \operatorname {S_{z}} \left(\theta \right)=\sum _{k=1}^{\infty }{\frac {\sin \left(k\cdot \theta \right)}{k^{z}}}}
James Whitbread Lee Glaisher ) und
C
z
(
θ
)
=
∑
k
=
1
∞
cos
(
k
⋅
θ
)
k
z
{\displaystyle \operatorname {C_{z}} \left(\theta \right)=\sum _{k=1}^{\infty }{\frac {\cos \left(k\cdot \theta \right)}{k^{z}}}}
Die Clausen-Funktion steht in Beziehung zum Polylogarithmus :
Cl
s
(
θ
)
=
Im
(
Li
s
(
e
i
θ
)
)
{\displaystyle \operatorname {Cl} _{s}(\theta )=\operatorname {Im} (\operatorname {Li} _{s}(e^{i\theta }))}
Ernst Kummer und Rogers führen folgende für
0
≤
θ
≤
2
π
{\displaystyle 0\leq \theta \leq 2\pi }
Li
2
(
e
i
θ
)
=
ζ
(
2
)
−
θ
(
2
π
−
θ
)
/
4
+
i
Cl
2
(
θ
)
{\displaystyle \operatorname {Li} _{2}(e^{i\theta })=\zeta (2)-\theta (2\pi -\theta )/4+i\operatorname {Cl} _{2}(\theta )}
Für rationale Werte von
θ
/
π
{\displaystyle \theta /\pi }
sin
(
n
θ
)
{\displaystyle \sin(n\theta )}
zyklischen Gruppe aufgefasst werden. Folglich kann
Cl
s
(
θ
)
{\displaystyle \operatorname {Cl} _{s}(\theta )}
hurwitzsche Zeta-Funktion beinhaltet. Das erlaubt es, Beziehungen zwischen bestimmten dirichletschen L-Funktionen einfach zu berechnen.
Die Clausen-Funktion kann auch als Methode betrachtet werden, um folgenden divergenten Fourier-Reihen eine Bedeutung zu geben:
sin
(
θ
)
+
2
sin
(
2
θ
)
+
3
sin
(
3
θ
)
+
…
{\displaystyle \sin(\theta )+2\sin(2\theta )+3\sin(3\theta )+\dots }
was mit
Cl
−
1
(
θ
)
{\displaystyle \operatorname {Cl} _{-1}(\theta )}
cos
(
θ
)
+
cos
(
2
θ
)
+
cos
(
3
θ
)
+
⋯
=
−
∫
d
θ
Cl
−
1
(
θ
)
{\displaystyle \cos(\theta )+\cos(2\theta )+\cos(3\theta )+\dots =-\int d{\theta }\operatorname {Cl} _{-1}(\theta )}
Dieses Ergebnis kann durch analytische Fortsetzung für alle negativen
s
{\displaystyle s}
Eine Reihenentwicklung für die Clausen-Funktion (für
|
θ
|
<
2
π
{\displaystyle |\theta |<2\pi }
Cl
2
(
θ
)
θ
=
1
−
log
|
θ
|
+
∑
n
=
1
∞
ζ
(
2
n
)
n
(
2
n
+
1
)
(
θ
2
π
)
2
n
.
{\displaystyle {\frac {\operatorname {Cl} _{2}(\theta )}{\theta }}=1-\log |\theta |+\sum _{n=1}^{\infty }{\frac {\zeta (2n)}{n(2n+1)}}\left({\frac {\theta }{2\pi }}\right)^{2n}{\text{.}}}
ζ
(
s
)
{\displaystyle \zeta (s)}
riemannsche Zeta-Funktion . Eine schneller konvergierende Reihe ist
Cl
2
(
θ
)
θ
=
3
−
log
[
|
θ
|
(
1
−
θ
2
4
π
2
)
]
−
2
π
θ
log
(
2
π
+
θ
2
π
−
θ
)
+
∑
n
=
1
∞
ζ
(
2
n
)
−
1
n
(
2
n
+
1
)
(
θ
2
π
)
2
n
.
{\displaystyle {\frac {\operatorname {Cl} _{2}(\theta )}{\theta }}=3-\log \left[|\theta |\left(1-{\frac {\theta ^{2}}{4\pi ^{2}}}\right)\right]-{\frac {2\pi }{\theta }}\log \left({\frac {2\pi +\theta }{2\pi -\theta }}\right)+\sum _{n=1}^{\infty }{\frac {\zeta (2n)-1}{n(2n+1)}}\left({\frac {\theta }{2\pi }}\right)^{2n}{\text{.}}}
Die Konvergenz wird dadurch sichergestellt, dass
ζ
(
n
)
−
1
{\displaystyle \zeta (n)-1}
n
{\displaystyle n}
Einige Spezialfälle sind gegeben durch:[ 2]
S
1
(
θ
)
=
1
2
⋅
π
−
1
2
⋅
θ
S
3
(
θ
)
=
1
6
⋅
π
2
⋅
θ
−
1
4
⋅
π
⋅
θ
2
+
1
12
⋅
θ
3
S
5
(
θ
)
=
1
90
⋅
π
4
⋅
θ
−
1
36
⋅
π
2
⋅
θ
3
+
1
48
⋅
π
⋅
θ
4
−
1
240
⋅
θ
5
C
2
(
θ
)
=
1
6
⋅
π
2
⋅
θ
−
1
2
⋅
π
⋅
θ
+
1
4
⋅
θ
2
C
4
(
θ
)
=
1
90
⋅
π
4
⋅
θ
−
1
12
⋅
π
2
⋅
θ
2
+
1
12
⋅
π
⋅
θ
3
−
1
48
⋅
θ
4
{\displaystyle {\begin{aligned}\operatorname {S_{1}} \left(\theta \right)&={\frac {1}{2}}\cdot \pi -{\frac {1}{2}}\cdot \theta \\\operatorname {S_{3}} \left(\theta \right)&={\frac {1}{6}}\cdot \pi ^{2}\cdot \theta -{\frac {1}{4}}\cdot \pi \cdot \theta ^{2}+{\frac {1}{12}}\cdot \theta ^{3}\\\operatorname {S_{5}} \left(\theta \right)&={\frac {1}{90}}\cdot \pi ^{4}\cdot \theta -{\frac {1}{36}}\cdot \pi ^{2}\cdot \theta ^{3}+{\frac {1}{48}}\cdot \pi \cdot \theta ^{4}-{\frac {1}{240}}\cdot \theta ^{5}\\\operatorname {C_{2}} \left(\theta \right)&={\frac {1}{6}}\cdot \pi ^{2}\cdot \theta -{\frac {1}{2}}\cdot \pi \cdot \theta +{\frac {1}{4}}\cdot \theta ^{2}\\\operatorname {C_{4}} \left(\theta \right)&={\frac {1}{90}}\cdot \pi ^{4}\cdot \theta -{\frac {1}{12}}\cdot \pi ^{2}\cdot \theta ^{2}+{\frac {1}{12}}\cdot \pi \cdot \theta ^{3}-{\frac {1}{48}}\cdot \theta ^{4}\\\end{aligned}}}
(für
0
≤
θ
≤
2
⋅
π
{\displaystyle 0\leq \theta \leq 2\cdot \pi }
Weitere Spezialfälle sind:
S
n
(
θ
)
=
i
2
⋅
[
L
i
n
(
exp
(
−
θ
⋅
i
)
)
−
L
i
n
(
exp
(
θ
⋅
i
)
)
]
C
n
(
θ
)
=
1
2
⋅
[
L
i
n
(
exp
(
−
θ
⋅
i
)
)
+
L
i
n
(
exp
(
θ
⋅
i
)
)
]
{\displaystyle {\begin{aligned}\operatorname {S_{n}} \left(\theta \right)&={\frac {i}{2}}\cdot \left[\operatorname {Li_{n}} \left(\exp \left(-\theta \cdot i\right)\right)-\operatorname {Li_{n}} \left(\exp \left(\theta \cdot i\right)\right)\right]\\\operatorname {C_{n}} \left(\theta \right)&={\frac {1}{2}}\cdot \left[\operatorname {Li_{n}} \left(\exp \left(-\theta \cdot i\right)\right)+\operatorname {Li_{n}} \left(\exp \left(\theta \cdot i\right)\right)\right]\\\end{aligned}}}
wobei
L
i
n
{\displaystyle \operatorname {Li_{n}} }
Polylogarithmus ist,
T
i
2
(
tan
(
θ
)
)
=
θ
⋅
log
(
tan
(
θ
)
)
+
1
2
⋅
C
l
2
(
2
⋅
θ
)
+
1
2
⋅
C
l
2
(
π
−
2
⋅
θ
)
{\displaystyle \operatorname {Ti_{2}} \left(\tan \left(\theta \right)\right)=\theta \cdot \log \left(\tan \left(\theta \right)\right)+{\frac {1}{2}}\cdot \operatorname {Cl_{2}} \left(2\cdot \theta \right)+{\frac {1}{2}}\cdot \operatorname {Cl_{2}} \left(\pi -2\cdot \theta \right)}
für
0
≤
tan
(
θ
)
≤
1
{\displaystyle 0\leq \tan \left(\theta \right)\leq 1}
T
i
2
{\displaystyle \operatorname {Ti_{2}} }
Arkustangensintegral ist,
Cl
2
(
2
π
z
)
=
2
π
log
(
G
(
1
−
z
)
G
(
1
+
z
)
)
+
2
π
z
log
(
π
sin
π
z
)
{\displaystyle \operatorname {Cl} _{2}(2\pi z)=2\pi \log \left({\frac {G(1-z)}{G(1+z)}}\right)+2\pi z\log \left({\frac {\pi }{\sin \pi z}}\right)}
Cl
2
(
2
π
z
)
=
2
π
log
(
G
(
1
−
z
)
G
(
z
)
)
−
2
π
log
Γ
(
z
)
+
2
π
z
log
(
π
sin
π
z
)
{\displaystyle \operatorname {Cl} _{2}(2\pi z)=2\pi \log \left({\frac {G(1-z)}{G(z)}}\right)-2\pi \log \Gamma (z)+2\pi z\log \left({\frac {\pi }{\sin \pi z}}\right)}
wobei
G
{\displaystyle G}
Γ
{\displaystyle \Gamma }
Gammafunktion ist,
Cl
2
(
θ
)
=
L
s
2
0
(
θ
)
{\displaystyle \operatorname {Cl} _{2}(\theta )={\mathcal {L}}s_{2}^{0}(\theta )}
[ 3]
wobei
L
s
2
0
{\displaystyle {\mathcal {L}}s_{2}^{0}}
L
s
n
m
(
θ
)
=
−
∫
0
θ
x
m
log
n
−
m
−
1
|
2
sin
x
2
|
d
x
{\displaystyle {\displaystyle {\mathcal {L}}s_{n}^{m}(\theta )=-\int _{0}^{\theta }x^{m}\log ^{n-m-1}\left|2\sin {\frac {x}{2}}\right|\,dx}}
Cl
s
(
π
2
)
=
β
(
s
)
{\displaystyle \operatorname {Cl} _{s}\left({\frac {\pi }{2}}\right)=\beta (s)}
wobei
β
(
s
)
{\displaystyle \beta (s)}
dirichletsche Beta-Funktion ist.
Einige spezielle Werte sind:
Cl
2
(
π
2
)
=
K
{\displaystyle \operatorname {Cl} _{2}\left({\frac {\pi }{2}}\right)=K}
Cl
2
(
π
3
)
=
3
π
log
(
G
(
2
3
)
G
(
1
3
)
)
−
3
π
log
Γ
(
1
3
)
+
π
log
(
2
π
3
)
{\displaystyle \operatorname {Cl} _{2}\left({\frac {\pi }{3}}\right)=3\pi \log \left({\frac {G\left({\frac {2}{3}}\right)}{G\left({\frac {1}{3}}\right)}}\right)-3\pi \log \Gamma \left({\frac {1}{3}}\right)+\pi \log \left({\frac {2\pi }{\sqrt {3}}}\right)}
Cl
2
(
2
π
3
)
=
2
π
log
(
G
(
2
3
)
G
(
1
3
)
)
−
2
π
log
Γ
(
1
3
)
+
2
π
3
log
(
2
π
3
)
{\displaystyle \operatorname {Cl} _{2}\left({\frac {2\pi }{3}}\right)=2\pi \log \left({\frac {G\left({\frac {2}{3}}\right)}{G\left({\frac {1}{3}}\right)}}\right)-2\pi \log \Gamma \left({\frac {1}{3}}\right)+{\frac {2\pi }{3}}\log \left({\frac {2\pi }{\sqrt {3}}}\right)}
Cl
2
(
π
4
)
=
2
π
log
(
G
(
7
8
)
G
(
1
8
)
)
−
2
π
log
Γ
(
1
8
)
+
π
4
log
(
2
π
2
−
2
)
{\displaystyle \operatorname {Cl} _{2}\left({\frac {\pi }{4}}\right)=2\pi \log \left({\frac {G\left({\frac {7}{8}}\right)}{G\left({\frac {1}{8}}\right)}}\right)-2\pi \log \Gamma \left({\frac {1}{8}}\right)+{\frac {\pi }{4}}\log \left({\frac {2\pi }{\sqrt {2-{\sqrt {2}}}}}\right)}
Cl
2
(
3
π
4
)
=
2
π
log
(
G
(
5
8
)
G
(
3
8
)
)
−
2
π
log
Γ
(
3
8
)
+
3
π
4
log
(
2
π
2
+
2
)
{\displaystyle \operatorname {Cl} _{2}\left({\frac {3\pi }{4}}\right)=2\pi \log \left({\frac {G\left({\frac {5}{8}}\right)}{G\left({\frac {3}{8}}\right)}}\right)-2\pi \log \Gamma \left({\frac {3}{8}}\right)+{\frac {3\pi }{4}}\log \left({\frac {2\pi }{\sqrt {2+{\sqrt {2}}}}}\right)}
Cl
2
(
π
6
)
=
2
π
log
(
G
(
11
12
)
G
(
1
12
)
)
−
2
π
log
Γ
(
1
12
)
+
π
6
log
(
2
π
2
3
−
1
)
{\displaystyle \operatorname {Cl} _{2}\left({\frac {\pi }{6}}\right)=2\pi \log \left({\frac {G\left({\frac {11}{12}}\right)}{G\left({\frac {1}{12}}\right)}}\right)-2\pi \log \Gamma \left({\frac {1}{12}}\right)+{\frac {\pi }{6}}\log \left({\frac {2\pi {\sqrt {2}}}{{\sqrt {3}}-1}}\right)}
Cl
2
(
5
π
6
)
=
2
π
log
(
G
(
7
12
)
G
(
5
12
)
)
−
2
π
log
Γ
(
5
12
)
+
5
π
6
log
(
2
π
2
3
+
1
)
{\displaystyle \operatorname {Cl} _{2}\left({\frac {5\pi }{6}}\right)=2\pi \log \left({\frac {G\left({\frac {7}{12}}\right)}{G\left({\frac {5}{12}}\right)}}\right)-2\pi \log \Gamma \left({\frac {5}{12}}\right)+{\frac {5\pi }{6}}\log \left({\frac {2\pi {\sqrt {2}}}{{\sqrt {3}}+1}}\right)}
wobei K die catalansche Konstante ist.
Leonard Lewin (Hrsg.): Structural Properties of Polylogarithms . American Mathematical Society, Providence (RI) 1991, ISBN 0-8218-4532-2 (englisch).
Jonathan M. Borwein, David M. Bradley, Richard E. Crandall: Computational Strategies for the Riemann Zeta Function . In: J. Comp. App. Math . Band 121 , 2000, S. 11 (englisch, maths.ex.ac.uk [PDF; 526 kB ]).
↑ Eric W. Weisstein: Clausen Function. Abgerufen am 15. Februar 2023 (englisch). ↑ Eric W. Weisstein: Clausen Function. Abgerufen am 15. Februar 2023 (englisch). ↑ Eric W. Weisstein: Log Sine Function. Abgerufen am 15. Februar 2023 (englisch). 
![{\displaystyle [0,\,2\cdot \pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3afbd57ef95482ee527ca7968eb9f97e6461ce9d)
![{\displaystyle {\frac {\operatorname {Cl} _{2}(\theta )}{\theta }}=3-\log \left[|\theta |\left(1-{\frac {\theta ^{2}}{4\pi ^{2}}}\right)\right]-{\frac {2\pi }{\theta }}\log \left({\frac {2\pi +\theta }{2\pi -\theta }}\right)+\sum _{n=1}^{\infty }{\frac {\zeta (2n)-1}{n(2n+1)}}\left({\frac {\theta }{2\pi }}\right)^{2n}{\text{.}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab4a7d9ba923c575e90a886e57796b714cc5f3d8)
![{\displaystyle {\begin{aligned}\operatorname {S_{n}} \left(\theta \right)&={\frac {i}{2}}\cdot \left[\operatorname {Li_{n}} \left(\exp \left(-\theta \cdot i\right)\right)-\operatorname {Li_{n}} \left(\exp \left(\theta \cdot i\right)\right)\right]\\\operatorname {C_{n}} \left(\theta \right)&={\frac {1}{2}}\cdot \left[\operatorname {Li_{n}} \left(\exp \left(-\theta \cdot i\right)\right)+\operatorname {Li_{n}} \left(\exp \left(\theta \cdot i\right)\right)\right]\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f876610d361ad8b4bcfa48133a18889abdbbbd1)
