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the beta coefficient, the non-diversifiable risk, of an asset in mathematical finance; the sideslip angle of an airplane; a beta particle (e − or e +) the beta brain wave in brain or cognitive sciences; ecliptic latitude in astronomy; the ratio of plasma pressure to magnetic pressure in plasma physics; β-reduction in lambda calculus
In probability theory and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distributions. [1] The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. [2] There are two equivalent parameterizations in common use:
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. It is defined by the integral
where () is the gamma function. In the special case that the four quantities n {\displaystyle n} , n + α {\displaystyle n+\alpha } , n + β {\displaystyle n+\beta } , n + α + β {\displaystyle n+\alpha +\beta } are nonnegative integers, the Jacobi polynomial can be written as
In trigonometry, Mollweide's formula is a pair of relationships between sides and angles in a triangle. [ 1 ] [ 2 ] A variant in more geometrical style was first published by Isaac Newton in 1707 and then by Friedrich Wilhelm von Oppel [ de ] in 1746.
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.
That β does indeed represent phase can be seen from Euler's formula: e i θ = cos θ + i sin θ {\displaystyle e^{i\theta }=\cos {\theta }+i\sin {\theta }\ } which is a sinusoid which varies in phase as θ varies but does not vary in amplitude because
If vectors u and v have direction cosines (α u, β u, γ u) and (α v, β v, γ v) respectively, with an angle θ between them, their units vectors are ^ = + + (+ +) = + + ^ = + + (+ +) = + +. Taking the dot product of these two unit vectors yield, ^ ^ = + + = , where θ is the angle between the two unit vectors, and is also the angle between u and v.