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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 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.
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:
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
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.
The natural logarithm of x is the power to which e would have to be raised to equal x. For example, ln 7.5 is 2.0149..., because e 2.0149... = 7.5. 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.
where C is the circumference of a circle, d is the diameter, and r is the radius.More generally, = where L and w are, respectively, the perimeter and the width of any curve of constant width.
Graph of the number of primes ending in 1, 3, 7, and 9 up to n for n < 10 000. Another example is the distribution of the last digit of prime numbers. Except for 2 and 5, all prime numbers end in 1, 3, 7, or 9. Dirichlet's theorem states that asymptotically, 25% of all primes end in each of these four digits.