Functions and Analysis

   

Ramanujan Value of Ln(x) When X Tends to Zero

Authors: Jesús Sánchez

As we know, the natural logarithm at zero diverges, towards minus infinity: lim┬(x→0)⁡〖Ln(x)〗=-∞ But, as happens with other functions or series that diverge at some points, it has a Ramanujan or Cauchy principal value (a finite value) associated to that point. In this case, it will be calculated to be: lim┬(x→0)⁡〖Ln(x)〗=-γ Being γ the Euler-Mascheroni constant 0.577215... It will be shown that Ln(0) tends to the negative of the sum of the harmonic series (that of course, diverges). But the harmonic series has a Cauchy principal value that is γ, the Euler-Mascheroni constant. So the finite associated value to Ln(0) will be calculated as - γ .

Comments: 3 Pages.

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Submission history

[v1] 2019-04-06 08:36:03

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