Difference between revisions of "Dilogarithm"

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=Properties=
 
=Properties=
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[[Relationship between Li 2(-1/x),Li 2(-x),Li 2(-1), and log^2(x)]]<br />
 
[[Relationship between Li 2(-1/x),Li 2(-x),Li 2(-1), and log^2(x)]]<br />
 
[[Derivative of Li 2(-1/x)]]<br />
 
[[Derivative of Li 2(-1/x)]]<br />
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[[Li2(z)=zPhi(z,2,1)]]<br />
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[[Li 2(z)=-Li 2(1/z)-(1/2)(log z)^2 + i pi log(z) + pi^2/3]]<br />
  
 
=References=
 
=References=
* {{BookReference|Polylogarithms and Associated Functions|1926|ed=2nd|edpage=Second Edition|Leonard Lewin|next=Taylor series of log(1-z)}}: (1.1) <br />
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* {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=findme|next=Relationship between dilogarithm and log(1-z)/z}}: $\S 1.11.1 (22)$
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* {{BookReference|Dilogarithms and Associated Functions|1958|Leonard Lewin|next=Taylor series of log(1-z)}}: $(1.1)$
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Li_2(z)+Li_2(1-z)=pi^2/6-log(z)log(1-z)}}: $27.7.2$ (<i>note: writes $\mathrm{Li}_2$ as $\sum_{k=1}^{\infty} \frac{(-1)^k(x-1)^k}{k^2}$ for $0 \leq x \leq 2$, equivalent to our definition on $\mathbb{R}$</i>)
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* {{BookReference|Polylogarithms and Associated Functions|1981|ed=2nd|edpage=Second Edition|Leonard Lewin|next=Taylor series of log(1-z)}}: $(1.1)$
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* {{BookReference|Structural Properties of Polylogarithms|1991|Leonard Lewin|next=Relationship between dilogarithm and log(1-z)/z}}: $(1.1)$
  
 
[http://authors.library.caltech.edu/43491/1/Volume%201.pdf (page 31)]<br />
 
[http://authors.library.caltech.edu/43491/1/Volume%201.pdf (page 31)]<br />
 
[http://maths.dur.ac.uk/~dma0hg/dilog.pdf The Dilogarithm function]<br />
 
[http://maths.dur.ac.uk/~dma0hg/dilog.pdf The Dilogarithm function]<br />
 
[http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-30308-4_1/fulltext.pdf]<br />
 
[http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-30308-4_1/fulltext.pdf]<br />
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{{:Logarithm and friends footer}}
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 23:22, 3 March 2018

The dilogarithm function $\mathrm{Li}_2$ is defined for $|z| \leq 1$ by $$\mathrm{Li}_2(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{z^k}{k^2},$$ which is a special case of the polylogarithm.

Properties

Relationship between dilogarithm and log(1-z)/z
Relationship between Li 2(1),Li 2(-1), and pi
Li 2(1)=pi^2/6
Relationship between Li 2(-1/x),Li 2(-x),Li 2(-1), and log^2(x)
Derivative of Li 2(-1/x)
Li2(z)=zPhi(z,2,1)
Li 2(z)=-Li 2(1/z)-(1/2)(log z)^2 + i pi log(z) + pi^2/3

References

(page 31)
The Dilogarithm function
[1]

Logarithm and friends