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Let λ 1, λ 2, and λ 3 be any three logarithms of algebraic numbers and γ be a non-zero algebraic number, and suppose that λ 1 λ 2 = γλ 3. Then λ 1 λ 2 = γλ 3 = 0. The exponential form of this conjecture is the following. Let x 1, x 2, and y be non-zero complex numbers and let γ be a non-zero algebraic number. Then at least one of ...
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, is taken to have the value
The sum of exponentials is a useful model in pharmacokinetics (chemical kinetics in general) for describing the concentration of a substance over time. The exponential terms correspond to first-order reactions, which in pharmacology corresponds to the number of modelled diffusion compartments. [2] [3]
Suppose we wish to generate random variables from Gamma(n + δ, 1), where n is a non-negative integer and 0 < δ < 1. Using the fact that a Gamma(1, 1) distribution is the same as an Exp(1) distribution, and noting the method of generating exponential variables, we conclude that if U is uniformly distributed on (0, 1], then −ln U is ...
The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles. The gamma function has no zeros, so the reciprocal gamma function 1 / Γ(z) is an entire function.
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
The binary number system expresses any number as a sum of powers of 2, and denotes it as a sequence of 0 and 1, separated by a binary point, where 1 indicates a power of 2 that appears in the sum; the exponent is determined by the place of this 1: the nonnegative exponents are the rank of the 1 on the left of the point (starting from 0), and ...
The natural sum is associative and commutative. It is always greater or equal to the usual sum, but it may be strictly greater. For example, the natural sum of ω and 1 is ω + 1 (the usual sum), but this is also the natural sum of 1 and ω. The natural product is associative and commutative and distributes over the natural sum.