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Hypergeometric0F1

Status: Stable

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

Description

Hypergeometric0F1[b, z]
    is the confluent hypergeometric 0F1, equal to HypergeometricPFQ[{}, {b}, z].

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Hypergeometric0F1[1/2, z]
Out[1]= Cosh[2 Sqrt[z]]

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]:= Hypergeometric0F1[1/2, z]
Out[1]= Cosh[2 Sqrt[z]]

With a different lower parameter the closed form switches to a hyperbolic sine — 0F1 is the engine behind the Bessel and circular/hyperbolic functions:

In[1]:= Hypergeometric0F1[3/2, z]
Out[1]= (1/2 Sinh[2 Sqrt[z]])/Sqrt[z]

Negative arguments give the trigonometric counterpart; with z = -Pi^2/16 the value of 0F1[3/2, z] is 2/Pi exactly, confirmed to 40 digits:

In[1]:= N[Hypergeometric0F1[3/2, -(Pi^2/16)]*Pi/2, 40]
Out[1]= 0.99999999999999999999999999999999999999991 + 2.3165577250480442772379354523138034341839e-41*I

Notes

Hypergeometric0F1[b, z] is the confluent limit HypergeometricPFQ[{}, {b}, z]. It converges for all z and underlies the Bessel functions: 0F1[1/2, z] = Cosh[2 Sqrt[z]] and 0F1[3/2, z] = Sinh[2 Sqrt[z]]/(2 Sqrt[z]). The tiny imaginary residue in the last result is numerical noise from the radical of a negative argument.