HypergeometricU

Tricomi's confluent hypergeometric function $U(a,b,z)$, for $b$ not an integer — see HypergeometricUStar for the $z^a$-regularized form Fungrim builds most of its identities from. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)FungrimHypergeometricU✓↗Fungrim entry6cf802↗9d3147✓↗c8fcc7✓↗Wolfram LanguageHypergeometricU!↗

HypergeometricU(a, b, z)Tricomi's confluent hypergeometric $U(a,b,z)$.

Typed signatures

Type	Package	Applies
(number, number, number) -> number	analytic	always

Domain: Special functions

Details

Tricomi's function $U(a,b,z)$ is the confluent hypergeometric equation's other standard solution, the one recessive as $z \to \infty$ where Kummer's $M(a,b,z)$ (${}_1F_1(a;b;z)$) grows. It specializes to elementary and other special functions at particular parameters — $U(a,a+1,z) = z^{-a}$ (DLMF 13.6.4), and $U(1,1,z) = e^z\,\Gamma(0,z) = e^z E_1(z)$, the exponential integral (DLMF 13.6.6 at $a=1$: $U(a,a,z) = e^z\,\Gamma(1-a,z)$; see Gamma).

• compute-engine 0.128 references this head only inside its identity rules (relating it to HypergeometricUStar) but never actually declares it as an operator, so there was nothing to extend — it is declared directly here, reusing the same Kummer connection-formula kernel as HypergeometricUStar.
• $b$ at (or very near) an integer is declined, for the same reason HypergeometricUStar declines there: the connection formula's $\Gamma(1-b)$ and $\Gamma(b-1)$ blow up, and the log-case limit that resolves it is not implemented. So is any point where the two terms of the formula cancel past a relative error of $10^{-11}$ (a $b$ within about $10^{-3}$ of an integer, or a large $z$): measured against mpmath on a grid of 546 real points, every answered point is within $5\times10^{-12}$.
• $U(a,b,z) \cdot z^a$ equals HypergeometricUStar(a,b,z) exactly (fungrim:c8fcc7), which is how this is cross-checked.

Examples

4 more kept as data, checked against the oracles

Basic #

a generic point, checked against mpmath's hyperu

live ↗

In$\mathrm{HypergeometricU}(1, 2.5, 3)$Out$0.382\,340\,662\,490\,315\,6$

Scope #

$b=1.001$ sits beside the integer $b=1$, where the connection formula's terms cancel: real, and good to about 11 digits (mpmath 0.7610286058670256)

live ↗

In$N(\mathrm{HypergeometricU}(\frac{1}{3}, 1.001, 2))$Out$0.761\,028\,605\,865\,931\,2$

Properties #

live ↗

In$N(\frac{1}{2}(\mathrm{HypergeometricU}(\frac{3}{2}, \frac{1}{2}, \frac{1}{2}))-\mathrm{HypergeometricU}(\frac{1}{2}, \frac{1}{2}, \frac{1}{2})+\mathrm{HypergeometricU}(-\frac{1}{2}, \frac{1}{2}, \frac{1}{2}))$Out$3.5\cdot10^{-15}$

Known value · Wolfram MathematicalFunctionData, HypergeometricU, FunctionalEquations

live ↗

In$N(\frac{1}{2}(\mathrm{HypergeometricU}(1/2, 3/2, 1/2))-\mathrm{HypergeometricU}(\frac{1}{2}, -\frac{1}{2}, \frac{1}{2}))$Out$6.5\cdot10^{-16}$

Known value · Wolfram MathematicalFunctionData, HypergeometricU, FunctionalEquations

live ↗

In$N(-\frac{1}{2}(5\mathrm{HypergeometricU}(\frac{1}{2}, \frac{1}{2}, 2))+\frac{1}{2}(\mathrm{HypergeometricU}(\frac{3}{2}, \frac{1}{2}, 2))+\mathrm{HypergeometricU}(-\frac{1}{2}, \frac{1}{2}, 2))$Out$1.265\cdot10^{-14}$

Known value · Wolfram MathematicalFunctionData, HypergeometricU, FunctionalEquations

live ↗

In$N(2\mathrm{HypergeometricU}(\frac{1}{2}, \frac{3}{2}, 2)-\frac{1}{2}(3\mathrm{HypergeometricU}(\frac{1}{2}, \frac{1}{2}, 2))-\mathrm{HypergeometricU}(\frac{1}{2}, -\frac{1}{2}, 2))$Out$4.645\cdot10^{-14}$

Known value · Wolfram MathematicalFunctionData, HypergeometricU, FunctionalEquations

Possible issues #

integer $b$ stays symbolic — the connection formula's log-case limit is not implemented

live ↗

In$\mathrm{HypergeometricU}(1, 2, 3)$Out$\mathrm{HypergeometricU}(1, 2, 3)$

At $b=1.00001$ the cancellation would cost more than 11 digits, so the kernel declines rather than answer wrong

live ↗

In$N(\mathrm{HypergeometricU}(\frac{1}{3}, 1.000\,01, 2))$Out$\mathrm{HypergeometricU}(0.\overline{3}, 1.000\,01, 2)$

About

Examples	12
Checked against	Wolfram Language 10 agree2 differ
Crosswalk	FungrimFungrim entry ×3Wolfram Language

Implementation

primitive · numericirreducibly numeric — it evaluates rather than rewrites

nativeenginetypescriptpackages/symbols/analysis/analytic/src/hypergeometric-ustar.ts

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

HypergeometricU[a,b,z]; mpmath.hyperu(a,b,z).

See also: HypergeometricUStar, Hypergeometric1F1Regularized