Special Functions

  • Mayer Humi
  • William Miller
Part of the Universitext book series (UTX)

Abstract

In the year 1812 Johann Frederick Carl Gauss published a comprehensive work studying the hypergeometric series which has the form
$$1+\frac{\alpha \beta}{\gamma}\ {x}+\frac{\alpha(1+\alpha)\beta(1+\beta)}{2!\gamma(1+\gamma)}{x}^{2}\ +\frac{\alpha(1+\alpha)(2+\alpha)\beta (1+\beta)(2+\beta)}{3!\gamma(1+\gamma)(2+\gamma)}{x}^{3}\ +\ ...$$
where α, β, γ, x are real numbers. The name hypergeometric comes from the fact that this series is a generalization of the geometric series. In fact, if we set α = 1 and β = γ in (1.1) above we have
$$1\ +{x}\ +\ {x}^{2}\ +\ {x}^{3}\ +\ ...$$
which is the well-known geometric series.

Keywords

Sine Cose sinO 

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Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Mayer Humi
    • 1
  • William Miller
    • 1
  1. 1.Department of Mathematical SciencesWorcester Polytechnic InstituteWorcesterUSA

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