Advanced theorems from complex analysis

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  1. Exchange of summation and integration
  2. Analyticity of a function defined by an integral
  3. Dirichlet series

Series Representations

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  1. Landau
  2. Hadamard
  3. Ramaswami

Functional equation

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  1. Euler discovery and partial proof
  2. Landau's proof
  3. Hardy's proof

Integral Representations

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A number of integral formulas involving the eta function can be listed.

  • A change of variable (Abel, 1823) in the integral representation of Euler's gamma fonction gives a Mellin transform which can be expressed in different ways. It is valid for  
 
  • Integration by parts yields formulas valid over successive unit strips in the left plane, starting with this one, for  
 
  • The next one is valid for  
 
 
  • In general, for  
 
  • This Lindelöf (1905) formula is valid over the whole complex plane, when the principal value is taken for the logarithm implicit in the exponential.
 

This corresponds to a Jensen (1895) formula for a related entire function, valid over the whole complex plane as proven by Lindelöf.