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  2. Generalized hypergeometric function - Wikipedia

    en.wikipedia.org/wiki/Generalized_hypergeometric...

    Plot of the generalized hypergeometric function pFq(a b z) with a=(2,4,6,8) and b=(2,3,5,7,11) in the complex plane from -2-2i to 2+2i created with Mathematica 13.1 function ComplexPlot3D. 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.

  3. Power series - Wikipedia

    en.wikipedia.org/wiki/Power_series

    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 ...

  4. List of mathematical series - Wikipedia

    en.wikipedia.org/wiki/List_of_mathematical_series

    An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.

  5. Geometric function theory - Wikipedia

    en.wikipedia.org/wiki/Geometric_function_theory

    Analytic continuation of natural logarithm (imaginary part) Analytic continuation is a technique to extend the domain of a given analytic function.Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.

  6. Function (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Function_(mathematics)

    Power series can be used to define functions on the domain in which they converge. For example, the exponential function is given by = =!. However, as the coefficients of a series are quite arbitrary, a function that is the sum of a convergent series is generally defined otherwise, and the sequence of the coefficients is the result of some ...

  7. Probability-generating function - Wikipedia

    en.wikipedia.org/.../Probability-generating_function

    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.

  8. Series expansion - Wikipedia

    en.wikipedia.org/wiki/Series_expansion

    A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.

  9. Generating function - Wikipedia

    en.wikipedia.org/wiki/Generating_function

    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.

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