Difference between revisions of "Relationship between dilogarithm and log(1-z)/z"
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==Theorem== | ==Theorem== | ||
The following formula holds: | The following formula holds: | ||
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==References== | ==References== | ||
| + | * {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=Dilogarithm|next=Li2(z)=zPhi(z,2,1)}}: $\S 1.11.1 (22)$ | ||
* {{BookReference|Dilogarithms and Associated Functions|1958|Leonard Lewin|prev=Taylor series of log(1-z)|next=findme}}: $(1.3)$ | * {{BookReference|Dilogarithms and Associated Functions|1958|Leonard Lewin|prev=Taylor series of log(1-z)|next=findme}}: $(1.3)$ | ||
| − | {{BookReference|Polylogarithms and Associated Functions|1981|ed=2nd|edpage=Second Edition|Leonard Lewin|prev=Taylor series of log(1-z)|next=findme}}: $(1.3)$ | + | *{{BookReference|Polylogarithms and Associated Functions|1981|ed=2nd|edpage=Second Edition|Leonard Lewin|prev=Taylor series of log(1-z)|next=findme}}: $(1.3)$ |
| + | * {{BookReference|Structural Properties of Polylogarithms|1991|Leonard Lewin|prev=Dilogarithm|next=findme}}: $(1.2)$ | ||
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 23:23, 3 March 2018
Theorem
The following formula holds: $$\mathrm{Li}_2(z)=-\displaystyle\int_0^z \dfrac{\log(1-z)}{z} \mathrm{d}z,$$ where $\mathrm{Li}_2$ denotes the dilogarithm and $\log$ denotes the logarithm.
Proof
References
- 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.11.1 (22)$
- 1958: Leonard Lewin: Dilogarithms and Associated Functions ... (previous) ... (next): $(1.3)$
- 1981: Leonard Lewin: Polylogarithms and Associated Functions (2nd ed.) ... (previous) ... (next): $(1.3)$
- 1991: Leonard Lewin: Structural Properties of Polylogarithms ... (previous) ... (next): $(1.2)$
