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Chebfun is a free/open-source software system written in MATLAB for numerical computation with functions of a real variable. It is based on the idea of overloading MATLAB's commands for vectors and matrices to analogous commands for functions and operators.
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:
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The Chebyshev functions, especially the second one ψ (x), are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, π (x) (see the exact formula below.) Both Chebyshev functions are asymptotic to x, a statement equivalent to the prime number theorem.
If p is a probability, then p/(1 − p) is the corresponding odds; the logit of the probability is the logarithm of the odds, i.e.: = = = = (). The base of the logarithm function used is of little importance in the present article, as long as it is greater than 1, but the natural logarithm with base e is the one most often used.
The elementary functions are constructed by composing arithmetic operations, the exponential function (), the natural logarithm (), trigonometric functions (,), and their inverses. The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's ...
Demonstrating log* 4 = 2 for the base-e iterated logarithm. The value of the iterated logarithm can be found by "zig-zagging" on the curve y = log b (x) from the input n, to the interval [0,1]. In this case, b = e. The zig-zagging entails starting from the point (n, 0) and iteratively moving to (n, log b (n) ), to (0, log b (n) ), to (log b (n ...
Any Taylor series for this function converges not only for x close enough to x 0 (as in the definition) but for all values of x (real or complex). The trigonometric functions, logarithm, and the power functions are analytic on any open set of their domain. Most special functions (at least in some range of the complex plane): hypergeometric ...