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In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy, [1] but remained relatively unknown until Hadamard rediscovered it. [2]
Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(X = i) in the probability mass function for a random variable X, and to make available the well-developed theory of power series with non-negative coefficients.
The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynomial with infinitely many terms. Conversely, every polynomial is a power ...
Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.
is a formal power series with rational coefficients a n, which has a non-zero radius of convergence in the complex plane, and within it represents an analytic function that is in fact an algebraic function. Then Eisenstein's theorem states that there exists a non-zero integer A, such that A n a n are all integers.
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function , which may then be defined over a wider domain of the argument by analytic continuation .
In order for the solution method to work, as in linear equations, it is necessary to express every term in the nonlinear equation as a power series so that all of the terms may be combined into one power series. As an example, consider the initial value problem ″ + ′ + ′ =; = , ′ = which describes a solution to capillary-driven flow in ...
The approximation ( +) and its equivalent form + ( + ( +)) can be obtained by rearranging Stirling's extended formula and observing a coincidence between the resultant power series and the Taylor series expansion of the hyperbolic sine function.