False Logarithmic Series


Euler (1738, 1753) considered the series


He showed that just like log_a(a^n)=n, s_a(a^n)=n for nonnegative integers n, though s_a(x) is a different function from log_a(x). s_a(a^x) (red) and log_a(a^x) (blue) for a=2, showing their coincidence at positive integers.

A closed form is given by


where psi_q(z) is the q-polygamma function.

See also

Logarithmic Series

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Euler, L. "Methodus generalis summandi progressiones." Commentarii academiae scientiarum imperialis Petropolitanae 6, pp. 68-97, (1732/33) 1738. Reprinted in Opera omnia I. 14, pp. 42-72.Euler, L. "Consideratio quarundam serierum quae singularibus proprietatibus sunt praeditae." Novi commentarii academiae scientiarum imperialis Petropolitanae 3, 10-12, (1750/51) 1753. Reprinted in Opera omnia I. 14, pp. 516-541.Sandifer, E. "How Euler Did It: A False Logarithmic Series." Dec. 2007.

Referenced on Wolfram|Alpha

False Logarithmic Series

Cite this as:

Weisstein, Eric W. "False Logarithmic Series." From MathWorld--A Wolfram Web Resource.

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