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The above expansion holds because the derivative of e x with respect to x is also e x, and e 0 equals 1. This leaves the terms ( x − 0) n in the numerator and n ! in the denominator of each term in the infinite sum.
Because log(x) is the sum of the terms of the form log(1 + 2 −k) corresponding to those k for which the factor 1 + 2 −k was included in the product P, log(x) may be computed by simple addition, using a table of log(1 + 2 −k) for all k. Any base may be used for the logarithm table. [53]
The correct second term of this expansion is 1 / 2n , where the given one works well to approximate roots with small n. Another improvement of Hermite's formula can be given: [ 11 ] x n = − n + 1 log n − 1 2 n ( log n ) 2 + O ( 1 n 2 ( log n ) 2 ) . {\displaystyle x_{n}=-n+{\frac {1}{\log n}}-{\frac {1}{2n(\log n)^{2}}}+O ...
The coefficients of the terms with k > 1 of z 1−k in the last expansion are simply where the B k are the Bernoulli numbers. The gamma function also has Stirling Series (derived by Charles Hermite in 1900) equal to [ 43 ] l o g Γ ( 1 + x ) = x ( x − 1 ) 2 ! log ( 2 ) + x ( x − 1 ) ( x − 2 ) 3 !
The natural logarithm of e itself, ln e, is 1, because e 1 = e, while the natural logarithm of 1 is 0, since e 0 = 1. The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/ x from 1 to a [ 4 ] (with the area being negative when 0 < a < 1 ).
The multiple valued version of log(z) is a set, but it is easier to write it without braces and using it in formulas follows obvious rules. log(z) is the set of complex numbers v which satisfy e v = z; arg(z) is the set of possible values of the arg function applied to z. When k is any integer:
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.