Dilogarithm

In mathematics , various special functions are called a dilogarithm . The classic dilogarithm is a special case of the polylogarithm .

Classic dilogarithm

Values ​​of the classical dilogarithm on the real axis. (The imaginary part is identically zero there.)

→ Main article : polylogarithm

The classic Dilogarithm for complex numbers with defined by the power series ${\displaystyle z}$ ${\displaystyle |z|<1}$

${\displaystyle \operatorname {Li} _{2}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{2}}}}$.

It can be continued through analytical continuation on : ${\displaystyle \mathbb {C} \setminus \left[1,\infty \right]}$

${\displaystyle \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\frac {\log(1-t)}{t}}\,\mathrm {d} t.}$

(This must be integrated along a path in .) ${\displaystyle \mathbb {C} \setminus \left[1,\infty \right]}$

Bloch-Wigner dilogarithm

The Bloch-Wigner dilogarithm is defined for by ${\displaystyle z\in \mathbb {C} }$

${\displaystyle \operatorname {D} _{2}(z)=\operatorname {Im} (\operatorname {Li} _{2}(z))+\arg(1-z)\log |z|}$.

It's well-defined and steady, even in . ${\displaystyle \left[1,\infty \right]}$

It is analytical in , in 0 and 1 it has singularities of type . ${\displaystyle \mathbb {C} \setminus \left\{0,1\right\}}$ ${\displaystyle r\log(r)}$

Rogers dilogarithm

The Rogers dilogarithm is defined by

${\displaystyle L(x)={\frac {6}{\pi ^{2}}}\left(\operatorname {Li} _{2}(x)+{\frac {1}{2}}\log(x)\log(1-x)\right)}$

for . ${\displaystyle 0<x<1}$

Another common definition is

${\displaystyle R(x)=\operatorname {Li} _{2}(x)+{\frac {1}{2}}\log(x)\log(1-x)-{\frac {\pi ^{2}}{6}}}$.

This depends on the former via

${\displaystyle R(x)={\frac {\pi ^{2}}{6}}(L(x)-1)}$

together.

One can (discontinuously) continue on completely through and ${\displaystyle R}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle R(1)=0,R(0)=-{\frac {\pi ^{2}}{6}}}$

${\displaystyle R(x)=\left\{{\begin{array}{c}-R(1/x)\ {\mbox{für}}\ x>1\\-R(x/(x-1))\ {\mbox{für}}\ x<0\end{array}}\right\}}$

Elliptical dilogarithm

Let be an elliptic curve defined over . It can be parameterized by means of a grid using the Weierstrasse schen function ${\displaystyle E}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \Lambda =\left\{1,\tau \right\}}$

${\displaystyle \mathbb {C} /\Lambda \rightarrow E(\mathbb {C} )}$

${\displaystyle u}$mod . ${\displaystyle \Lambda \mapsto (p(u),p^{\prime }(u))}$

The elliptical dilogarithm is then defined by ${\displaystyle D^{E}\colon E(\mathbb {C} )\rightarrow \mathbb {C} }$

${\displaystyle D^{E}(p(u),p^{\prime }(u))=\sum _{n=-\infty }^{\infty }D_{2}(e^{2\pi i(n\tau +u)})}$,

where denotes the Bloch-Wigner dilogarithm. ${\displaystyle D_{2}}$

The elliptical dilogarithm agrees with the value of the L-function except for rational multiples of . ${\displaystyle \pi }$ ${\displaystyle L(E,2)}$

Special values

Classic dilogarithm

For the following numbers, and can be represented in closed form: ${\displaystyle z}$ ${\displaystyle \operatorname {Li} _{2}(z)}$

${\displaystyle \operatorname {Li} _{2}(-1)=-{\frac {{\pi }^{2}}{12}},\qquad \operatorname {Li} _{2}(0)=0}$,

${\displaystyle \operatorname {Li} _{2}\left({\frac {1}{2}}\right)={\frac {{\pi }^{2}}{12}}-{\frac {\ln ^{2}2}{2}},\qquad \operatorname {Li} _{2}(1)={\frac {{\pi }^{2}}{6}}}$,

${\displaystyle \operatorname {Li} _{2}\left(-{\frac {{\sqrt {5}}-1}{2}}\right)=-{\frac {{\pi }^{2}}{15}}+{\frac {1}{2}}\ln ^{2}{\frac {{\sqrt {5}}-1}{2}},\qquad \operatorname {Li} _{2}\left(-{\frac {{\sqrt {5}}+1}{2}}\right)=-{\frac {{\pi }^{2}}{10}}-\ln ^{2}{\frac {{\sqrt {5}}+1}{2}}}$,

${\displaystyle \operatorname {Li} _{2}\left({\frac {3+{\sqrt {5}}}{2}}\right)={\frac {{\pi }^{2}}{15}}-{\frac {1}{2}}\ln ^{2}{\frac {{\sqrt {5}}-1}{2}},\qquad \operatorname {Li} _{2}\left({\frac {{\sqrt {5}}+1}{2}}\right)={\frac {{\pi }^{2}}{10}}-\ln ^{2}{\frac {{\sqrt {5}}-1}{2}}}$.

With the sixth root of unity and the Gieseking constant one also has ${\displaystyle \omega ={\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i}$ ${\displaystyle V_{0}=1{,}0149\ldots }$

${\displaystyle \operatorname {Li} _{2}(\omega )={\frac {\pi ^{2}}{36}}+V_{0}i,\qquad \operatorname {Li} _{2}(\omega ^{2})=-{\frac {\pi ^{2}}{18}}+{\frac {2}{3}}V_{0}i}$

${\displaystyle \operatorname {Li} _{2}(1+\omega )={\frac {\pi ^{2}}{9}}+{\frac {2}{3}}(V_{0}+{\frac {1}{2}}\ln(3)\pi )i,\qquad \operatorname {Li} _{2}({\frac {1}{1+\omega }})={\frac {5\pi ^{2}}{72}}-{\frac {1}{8}}\ln(3)+({\frac {1}{12}}\ln(3)\pi -{\frac {2}{3}}V_{0})i}$.

Bloch-Wigner dilogarithm

So far, values ​​of the Bloch-Wigner dilogarithm can only be calculated numerically and only a few algebraic relations are known between values ​​of the Bloch-Wigner dilogarithm. An assumption by John Milnor states that : ${\displaystyle N\geq 3}$

the numbers for and are linearly independent over . ${\displaystyle D_{2}(e^{2\pi i{\frac {j}{N}}})}$ ${\displaystyle 0<j<{\frac {N}{2}}}$ ${\displaystyle \operatorname {ggT} (j,N)=1}$ ${\displaystyle \mathbb {Q} }$

Rogers dilogarithm

There are numerous algebraic identities between values ​​of in rational or algebraic arguments. Examples of special values ​​are ${\displaystyle L}$

${\displaystyle L(0)=0,\quad L\left({\frac {1}{2}}\right)={\frac {1}{2}},\quad L(1)=1,\quad L\left({\frac {3-{\sqrt {5}}}{2}}\right)={\frac {2}{5}},\quad L\left({\frac {{\sqrt {5}}-1}{2}}\right)={\frac {3}{5}}}$.

With the sixth root of unity and the Gieseking constant one has ${\displaystyle \omega ={\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i}$ ${\displaystyle V_{0}=1,0149...}$

${\displaystyle R(\omega )=-{\frac {\pi ^{2}}{12}}+V_{0}i,\qquad R(\omega ^{2})=-{\frac {\pi ^{2}}{6}}+({\frac {2}{3}}V_{0}+{\frac {1}{6}}\ln(3)\pi )i}$

${\displaystyle R(1+\omega )=({\frac {2}{3}}V_{0}+{\frac {1}{6}}\ln(3)\pi )i,\qquad R({\frac {1}{1+\omega }})=-{\frac {\pi ^{2}}{12}}-{\frac {1}{8}}\ln(3)+{\frac {1}{8}}(\ln(3))^{2}+(-{\frac {2}{3}}V_{0}+{\frac {1}{12}}\ln(3)\pi )i}$

Functional equations

Classic dilogarithm

The classical dilogarithm suffices for numerous functional equations, for example

${\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}(-z)={\frac {1}{2}}\operatorname {Li} _{2}(z^{2})}$

${\displaystyle \operatorname {Li} _{2}(1-z)+\operatorname {Li} _{2}\left(1-{\frac {1}{z}}\right)=-{\frac {\ln ^{2}z}{2}}}$

${\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}(1-z)={\frac {{\pi }^{2}}{6}}-\ln z\cdot \ln(1-z)}$

${\displaystyle \operatorname {Li} _{2}(-z)-\operatorname {Li} _{2}(1-z)+{\frac {1}{2}}\operatorname {Li} _{2}(1-z^{2})=-{\frac {{\pi }^{2}}{12}}-\ln z\cdot \ln(z+1)}$

${\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}\left({\frac {1}{z}}\right)=-{\frac {\pi ^{2}}{6}}-{\frac {1}{2}}\ln ^{2}(-z)}$

Bloch-Wigner dilogarithm

The Bloch-Wigner dilogarithm is sufficient for identities

${\displaystyle \operatorname {D} _{2}(z)=\operatorname {D} _{2}\left(1-{\frac {1}{z}}\right)=\operatorname {D} _{2}\left({\frac {1}{1-z}}\right)=-\operatorname {D} _{2}\left({\frac {1}{z}}\right)=-\operatorname {D} _{2}(1-z)=-\operatorname {D} _{2}\left({\frac {-z}{1-z}}\right)}$

and the 5-term relation

${\displaystyle \operatorname {D} _{2}(x)+\operatorname {D} _{2}(y)+\operatorname {D} _{2}\left({\frac {1-x}{1-xy}}\right)+\operatorname {D} _{2}(1-xy)+\operatorname {D} _{2}\left({\frac {1-y}{1-xy}}\right)=0.}$

Rogers dilogarithm

The Rogers dilogarithm fulfills the relationship

${\displaystyle L(x)+L(1-x)=1}$

and Abel's functional equation

${\displaystyle L(x)+L(y)=L(xy)+L\left({\frac {x(1-y)}{1-xy}}\right)+L\left({\frac {y(1-x)}{1-xy}}\right)}$.

For one has ${\displaystyle R}$

${\displaystyle R(x)+R(1-x)=-{\frac {\pi ^{2}}{6}}}$

and the 5-term relation

${\displaystyle R(x)-R(y)+R({\frac {y}{x}})-R({\frac {1-x^{-1}}{1-y^{-1}}})+R({\frac {1-x}{1-y}})=0}$,

in particular is a well-defined function on the Bloch group . ${\displaystyle R}$

See also

• Quantum logarithm

Web links

• Eric W. Weisstein : Dilogarithm . In: MathWorld (English).
• Don Zagier: The dilogarithm function (PDF; 1.6 MB)
• Matilde Lalin: The dilogarithm
• Dupont, Zickert: A dilogarithmic formula for the Cheeger-Chern-Simons class

Individual evidence

1. ^ K ₂ and L-functions of elliptic curves