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The logarithm of the gamma function satisfies the following formula due to Lerch: = ′ (,) ′ (), where is the Hurwitz zeta function, is the Riemann zeta function and the prime (′) denotes differentiation in the first variable.
Contour plot of the beta function. In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients.
Instead, the half-life is defined in terms of probability: "Half-life is the time required for exactly half of the entities to decay on average". In other words, the probability of a radioactive atom decaying within its half-life is 50%. [2] For example, the accompanying image is a simulation of many identical atoms undergoing radioactive decay.
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718 281 828 459. [1] 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 .
For each nonzero complex number , the principal value is the logarithm whose imaginary part lies in the interval (,]. [2] The expression Log 0 {\displaystyle \operatorname {Log} 0} is left undefined since there is no complex number w {\displaystyle w} satisfying e w = 0 {\displaystyle e^{w}=0} .
Overlaid graphs of the logarithm for bases 1 / 2 , 2, and e. Among all choices for the base, three are particularly common. These are b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and b = 2 (the binary logarithm). In mathematical analysis, the logarithm base e is widespread because of analytical properties ...
SymPy is an open-source Python library for symbolic computation.It provides computer algebra capabilities either as a standalone application, as a library to other applications, or live on the web as SymPy Live [2] or SymPy Gamma. [3]