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The dilogarithm along the real axis. In mathematics, the dilogarithm (or Spence's function), denoted as Li 2 (z), is a particular case of the polylogarithm.Two related special functions are referred to as Spence's function, the dilogarithm itself:
The logarithm of a complex number is thus a multi-valued function, because φ is multi-valued. Finally, the other exponential law ( e a ) k = e a k , {\displaystyle \left(e^{a}\right)^{k}=e^{ak},} which can be seen to hold for all integers k , together with Euler's formula, implies several trigonometric identities , as well as de Moivre's formula .
The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the p-th power of a number is p times the logarithm of the number itself; the logarithm of a p-th root is the logarithm of the number divided by p. The following ...
The logarithm in the table, however, is of that sine value divided by 10,000,000. [1]: p. 19 The logarithm is again presented as an integer with an implied denominator of 10,000,000. The table consists of 45 pairs of facing pages. Each pair is labeled at the top with an angle, from 0 to 44 degrees, and at the bottom from 90 to 45 degrees.
The full formula, together with precise estimates of its error, can be derived as follows. Instead of approximating !, one considers its natural logarithm, as this is a slowly varying function: (!) = + + + .
Alexander John Thompson (1885 in Plaistow, Essex – 17 June 1968 in Wallington, Surrey) is the author of the last great table of logarithms, published in 1952.This table, the Logarithmetica britannica gives the logarithms of all numbers from 1 to 100000 to 20 places and supersedes all previous tables of similar scope, in particular the tables of Henry Briggs, Adriaan Vlacq and Gaspard de Prony.
Microsoft founder Bill Gates is telling his “origin story” in his own words with the memoir Source Code, being released on Feb. 4 "My parents and early friends put me in a position to have a ...
Analytic continuation of natural logarithm (imaginary part) Suppose f is an analytic function defined on a non-empty open subset U of the complex plane C {\displaystyle \mathbb {C} } . If V is a larger open subset of C {\displaystyle \mathbb {C} } , containing U , and F is an analytic function defined on V such that