Can anyone solve this complicated integral with limited conditions?

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The integral ∫ {[ln(v)]^(s-1) - [ln(v)]^(-s)}/(v-1) dv, with limits from v=1 to v=e and 0 PREREQUISITES

  • Understanding of complex analysis, particularly integration techniques.
  • Familiarity with logarithmic functions and their properties.
  • Experience using Mathematica for symbolic computation.
  • Knowledge of convergence criteria for integrals.
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  • Research methods for evaluating complex integrals, specifically using residue theory.
  • Learn about convergence tests for improper integrals.
  • Explore advanced features of Mathematica for handling non-convergent integrals.
  • Study the properties of logarithmic functions in the context of complex variables.
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Mathematicians, students of complex analysis, and anyone involved in advanced calculus or integral evaluation will benefit from this discussion.

rman144
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I've looked everywhere for a method or approach to break down this integral, but so far, nothing. If anyone has any ideas or answers, I would be incredibly thankful:


∫ {[ln(v)]^(s-1) - [ln(v)]^(-s)}/(v-1) dv, with limits from v=1 to v=e, and 0<Re(s)<1

I've tried breaking it down to the real and imaginary parts, but even then I was back to the problem of trying to integrate with cos(u)+isin(u) issues.
 
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[tex]\int_1^e \frac{(\ln v)^{s - 1} - (\ln v)^{-s}}{v - 1} \,dv<br /> <br /> According to Mathematica, the integral doesn't converge. It doesn't seem to be able to find an indefinite integral either.[/tex]
 

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