U(a,b,z) returns results different from Mathematica/MATLAB when z is negative

[Haonan-Zhou]

Hi,

Thanks a lot for the wonderful package!

This issue might be related to #61 , but I tried to compare the result of the Tricomi confluent hypergeometric function HypergeometricFunctions.U(a,b,z) across softwares when the final argument z is a negative number. Here are some results:

Experiment 1: Evaluate U(0.5, 0.5, -2.5):

• Mathematica: HypergeometricU[0.5, 0.5, -2.5] yields 0.145492 - 0.810467i.
• MATLAB: kummerU(0.5, 0.5, -2.5) yields 0.1455 - 0.8105i.
• Julia: HypergeometricFunctions.U(0.5, 0.5, Complex(-2.5)) yields 0.0 - 1.1612906236707266im.

Note that HypergeometricFunctions.U(0.5, 0.5, -2.5) throws a DomainError with -2.5: Exponentiation yielding a complex result requires a complex argument. Probably one needs to make sure z^(1-b) is defined for negative real z.

Experiment 2: Evaluate U(-0.5 -0.5, -2.5):

• Mathematica: HypergeometricU[-0.5, -0.5, -2.5] yields 0.0727459 + 1.17591i.
• MATLAB: kummerU(-0.5, -0.5, -2.5) yields 0.0727 + 1.1759i.
• Julia: HypergeometricFunctions.U(-0.5, -0.5, Complex(-2.5)) yields 0.0 + 0.0im.

The issue does not seem to appear (the results matches across platforms) if z is a positive real or z is a complex number with a negative real part:

Experiment 3: Evaluate U(-0.5 -0.5, -2.5-1.5im):

• Mathematica: HypergeometricU[-0.5, -0.5, -2.5-1.5i] yields 0.595921 - 1.34976i.
• MATLAB: result is 0.5959 - 1.3498i.
• Julia: result is 0.5959210215481137 - 1.3497630048128746im.

Environment: Julia 1.8.5, package ver. 0.3.15. MATLAB R2022a. Mathematica 12.3.1.0 on a macOS 13.3.

[MikaelSlevinsky]

Thanks for filing the issue! Evaluating U near the branch cut (-∞,0] requires a different approach, such as Kummer's transformation #61.