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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 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 x is generally written as ln x, log e x, or sometimes, if the base e is implicit, simply log x. [2] [3] Parentheses are sometimes added for clarity, giving ln(x), log e (x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.
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 ...
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.
Going down from x + 1 to x, ψ decreases by 1 / x , ln(x − 1 / 2 ) decreases by ln(x + 1 / 2 ) / (x − 1 / 2 ), which is more than 1 / x , and ln x decreases by ln(1 + 1 / x ), which is less than 1 / x . From this we see that for any positive x greater than 1 / 2 ,
Plot of the logarithmic integral function li(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D. In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number theoretic significance.