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ln(r) is the standard natural logarithm of the real number r. Arg(z) is the principal value of the arg function; its value is restricted to (−π, π]. It can be computed using Arg(x + iy) = atan2(y, x). Log(z) is the principal value of the complex logarithm function and has imaginary part in the range (−π, π].
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. 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 n th partial sum can approximate ln(z) with arbitrary precision, provided the number of summands n is large enough. In elementary calculus, the series is said to converge to the function ln(z), and the function is the limit of the series. It is the Taylor series of the natural logarithm at z = 1.
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These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
The calculation of the Meissel–Mertens constant. [30] Lower bounds to specific prime gaps. [31] An approximation of the average number of divisors of all numbers from 1 to a given n. [32] The Lenstra–Pomerance–Wagstaff conjecture on the frequency of Mersenne primes. [33] An estimation of the efficiency of the euclidean algorithm. [34]