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logarithmic integral

Defines: 
logarithmic integral, logarithmus integralis, Eulerian logarithmic integral
Synonym: 
Li
Type of Math Object: 
Definition
Major Section: 
Reference
Groups audience: 

Mathematics Subject Classification

30E20 no label found33E20 no label found26A36 no label found

Comments

Hi, I am unsure whether z = 1 is a pole or an essential singularity of the logarithmic integral. It is clear that z = 1 is a pole of order 1 of 1/ln{z}.
Jussi

Neither --- it is a branch point. When one integrates a simple pole,
one genrates a logarithm. you can think of it this way. Write
1/ln{z} as 1/(z-1) + f(z) where f is analytic at z = 1 (in other
words, separate out the pole). Integrating, we find that li{z} =
log (z-1) + F(z), where F is the antiderivative of f, which makes it
analytic at z = 1.

Thank you, Ray! Thus I must remove the attribute "meromorphic" from the entry.
Jussi

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