# Incomplete Gamma Function with Continued Fractions

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Continued fractions provide a very effective toolset for approximating functions. Usually the continued fraction expansion of a function approximates the function better than its Taylor or Fourier series. This Demonstration compares the quality of three approximations to the incomplete gamma function . One is the Taylor series and the other two are continued fraction expansions. The first continued fraction expansion can be obtained as a canonical even contraction of a continued fraction using Euler's method to transform a series to an S-fraction. The other is a continued fraction expansion the author has developed as a canonical even contraction from the first one.

Contributed by: Andreas Lauschke (March 2011)
Open content licensed under CC BY-NC-SA

## Details

This Demonstration shows an example of the Taylor series providing a better approximation to a function in a certain range and a continued fraction approximation providing a better approximation in another range. It is generally a shortcoming of polynomials that for larger absolute values of they cannot approximate functions well that converge towards constants or do not have zeroes, as polynomials tend to ±∞ for large absolute . Rational function approximations—for example, continued fractions or Padé approximations—or certain special functions are much better. On the other hand, as can be seen here, the series approximates the incomplete Gamma function better for smaller values of as both continued fraction expansions must go through the origin, but . The optimal "change-over point" varies with the parameter and the number of terms used in the expansions.