Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 EulerFalseLog

Source Notebook

Evaluate Euler's false logarithmic series

Contributed by: Jan Mangaldan

ResourceFunction["EulerFalseLog"][a,z]

gives the false logarithmic series sa(z) of Euler.

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
ResourceFunction["EulerFalseLog"] can be evaluated to arbitrary numerical precision.
ResourceFunction["EulerFalseLog"] automatically threads over lists.

Examples

Basic Examples (2) 

Evaluate numerically:

In[1]:=
ResourceFunction["EulerFalseLog"][2., 3]
Out[1]=

Plot over a subset of the reals:

In[2]:=
Plot[ResourceFunction["EulerFalseLog"][2, x], {x, 0, 5}]
Out[2]=

Scope (4) 

Evaluate for complex arguments:

In[3]:=
ResourceFunction["EulerFalseLog"][1.3 - 2.4 I, 2.5 + I]
Out[3]=

Evaluate to high precision:

In[4]:=
N[ResourceFunction["EulerFalseLog"][E, 5/2], 50]
Out[4]=

The precision of the output tracks the precision of the input:

In[5]:=
ResourceFunction[ "EulerFalseLog"][E, 2.50000000000000000000000000000000000000000000]
Out[5]=

EulerFalseLog threads elementwise over lists:

In[6]:=
ResourceFunction["EulerFalseLog"][E, {1.3, 1.5, 1.7}]
Out[6]=

Applications (1) 

Visualize the difference between Euler's false logarithm and the logarithm:

In[7]:=
Manipulate[ Plot[{ResourceFunction["EulerFalseLog"][a, a^x], Log[a, a^x]}, {x, -2.1, 5}, {Epilog -> { RGBColor[1, 0, 0], PointSize[0.01], Point[{{-2, -2}, {-1, -1}, {0, 0}, {1, 1}, {2, 2}, {3, 3}, {4, 4}, {5, 5}}]}, PlotRange -> {-2.2, 5.5}}], {a, 2, 10, 1, Appearance -> "Labeled"}, SaveDefinitions -> True]
Out[7]=

Properties and Relations (1) 

EulerFalseLog can be expressed in terms of QPolyGamma for x=0:

In[8]:=
ResourceFunction["EulerFalseLog"][a, 0] == QPolyGamma[1, 1/a]/Log[a] + Log[a, a - 1] - 1 /. a -> N[E, 20]
Out[8]=

Version History

  • 1.0.0 – 15 January 2021

Source Metadata

  • Citation:
    • Gautschi W., On Euler’s attempt to compute logarithms by interpolation: A commentary to his letter of February 17, 1734, to Daniel Bernoulli. Journal of Computational and Applied Mathematics, vol. 219, no. 2, 408-415, 2008
      DOI: 10.1016/j.cam.2006.11.027
    • Koelink E., Van Assche W., Leonhard Euler and a q-analogue of the logarithm. Proceedings of the American Mathematical Society, vol. 137, 1663-1676, 2009
      DOI: 10.1090/S0002-9939-08-09374-X
    • Sandifer E., A false logarithm series. In How Euler Did It. MAA Online
      Available online http://eulerarchive.maa.org/hedi/HEDI-2007-12.pdf

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