Difference between revisions of "Lerch transcendent"

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The Lerch transcendent $\Phi$ is defined by
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The Lerch transcendent $\Phi$ is defined for $|z|<1$ and $a \in \mathbb{C} \setminus \{ 0,-1,-2,\ldots\}$ by
 
$$\Phi(z,s,a)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(a+k)^s}.$$
 
$$\Phi(z,s,a)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(a+k)^s}.$$
  
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[[Relationship between Lerch transcendent and Lerch zeta]]<br />
 
[[Relationship between Lerch transcendent and Lerch zeta]]<br />
 
[[Dirichlet beta in terms of Lerch transcendent]]<br />
 
[[Dirichlet beta in terms of Lerch transcendent]]<br />
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[[Legendre chi in terms of Lerch transcendent]]<br />
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[[Li2(z)=zPhi(z,2,1)]]<br />
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=References=
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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=findme}}: $\S 1.11 (1)$
  
 
[[Category:SpecialFunction]]
 
[[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