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The characteristic function is a way to describe a random variable X.The characteristic function, = [],a function of t, determines the behavior and properties of the probability distribution of X.
The arithmetic–geometric mean of two numbers, a 0 and b 0, is found by calculating the limit of the sequences + = +, + =, which both converge to the same limit. If = and = then the limit is () where () is the complete elliptic integral of the first kind
Using the P function mentioned above, the simplest known formula for π is for s = 1, but m > 1. Many now-discovered formulae are known for b as an exponent of 2 or 3 and m as an exponent of 2 or it some other factor-rich value, but where several of the terms of sequence A are zero.
The standard deviation of the distribution is (sigma). A random variable with a Gaussian distribution is said to be normally distributed , and is called a normal deviate . Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not ...
In mathematics, the Leibniz formula for π, named after Gottfried Wilhelm Leibniz, states that = + + = = +,. an alternating series.. It is sometimes called the Madhava–Leibniz series as it was first discovered by the Indian mathematician Madhava of Sangamagrama or his followers in the 14th–15th century (see Madhava series), [1] and was later independently rediscovered by James Gregory in ...
In mathematics, a Ramanujan–Sato series [1] [2] generalizes Ramanujan’s pi formulas such as, = = ()!! + to the form = = + by using other well-defined sequences of integers obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients (), and ,, employing modular forms of higher levels.
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Erich Hückel proposed a representation of the double bond as a combination of a sigma bond plus a pi bond. [ 7 ] [ 8 ] [ 9 ] The σ-π representation is the better-known one, and it is the one found in most textbooks since the late-20th century.