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Hypergeometric1F1

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Hypergeometric1F1[a, b, z]
    is Kummer's confluent hypergeometric 1F1, equal to HypergeometricPFQ[{a}, {b}, z].

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Hypergeometric1F1[a, b, 0]
Out[1]= 1

Implementation notes

Attributes: NumericFunction, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Hypergeometric1F1[1, 2, z]
Out[1]= (-1 + E^z)/z

A non-positive integer upper parameter terminates Kummer's series to a polynomial (here a scaled Laguerre polynomial):

In[1]:= Hypergeometric1F1[-2, 1, z]
Out[1]= 1 - 2 z + 1/2 z^2

The numeric path agrees with the closed form to full precision — evaluating 1F1[1, 2, 3] and (E^3 - 1)/3 to 40 digits:

In[1]:= N[Hypergeometric1F1[1, 2, 3], 40]
Out[1]= 6.3618456410625559136428432181939059656621

In[2]:= N[(E^3 - 1)/3, 40]
Out[2]= 6.3618456410625559136428432181939059656621

Notes

Hypergeometric1F1[a, b, z] is Kummer's confluent hypergeometric function, equal to HypergeometricPFQ[{a}, {b}, z], and converges for all z. A non-positive integer a truncates the series to a polynomial (the Laguerre/Hermite family); otherwise the function evaluates numerically at machine, MPFR, and complex precision.