Difference between revisions of "Lerch transcendent"
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| − | The Lerch transcendent $\Phi$ is defined by | + | The Lerch transcendent $\Phi$ is defined for $|z|<1$ and $a \in \mathbb{C} \setminus \{ 0,-1,-2,\ldots\}$ by |
$$\Phi(z,s,a)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(a+k)^s}.$$ | $$\Phi(z,s,a)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(a+k)^s}.$$ | ||
=Properties= | =Properties= | ||
| − | + | [[Lerch transcendent polylogarithm]]<br /> | |
| − | + | [[Relationship between Lerch transcendent and Lerch zeta]]<br /> | |
| + | [[Dirichlet beta in terms of Lerch transcendent]]<br /> | ||
| + | [[Legendre chi in terms of Lerch transcendent]]<br /> | ||
| + | [[Li2(z)=zPhi(z,2,1)]]<br /> | ||
| + | |||
| + | =References= | ||
| + | * {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=findme|next=findme}}: $\S 1.11 (1)$ | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 23:22, 3 March 2018
The Lerch transcendent $\Phi$ is defined for $|z|<1$ and $a \in \mathbb{C} \setminus \{ 0,-1,-2,\ldots\}$ by $$\Phi(z,s,a)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(a+k)^s}.$$
Properties
Lerch transcendent polylogarithm
Relationship between Lerch transcendent and Lerch zeta
Dirichlet beta in terms of Lerch transcendent
Legendre chi in terms of Lerch transcendent
Li2(z)=zPhi(z,2,1)
References
- 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.11 (1)$
