Hypergeometric1F1Regularized

The regularized Kummer confluent hypergeometric function ${}_1F_1(a,b;z) / \Gamma(b)$ — entire in $b$ and $z$ . Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)FungrimHypergeometric1F1Regularized✓Fungrim entrya047eb✓f7f84e✓Wolfram LanguageHypergeometric1F1Regularized✓

Hypergeometric1F1Regularized(a, b, z)

{}_1F_1(a,b;z) / \Gamma(b)

.

Typed signatures

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

Domain: Special functions

Details

compute-engine declares Hypergeometric1F1 itself (real and complex $z$ ) but not this regularized form. Computed by the same $1/\Gamma$ -per-term series as Hypergeometric0F1Regularized, so it stays finite at $b$ a nonpositive integer rather than dividing by Gamma(b)'s pole there.
Entire in $z$ ( $p = q$ for this series), so never declines on $z$ .
N(x, d) past a double's digits runs a BigDecimal series for real arguments, to $d$ digits; a complex argument stays unevaluated there rather than print a double's digits as if they were $d$ .

Examples

6 more kept as data, checked against the oracles

Basic #

matches Hypergeometric1F1(1, 2, 0.5) / Gamma(2)

In

\mathrm{Hypergeometric1F1Regularized}(1, 2, 0.5)

Out

1.297\,442\,541\,400\,255\,5

$b = -1$ is a pole of $\Gamma$ , but the regularized form is finite there

In

\mathrm{Hypergeometric1F1Regularized}(1, -1, 0.7)

Out

0.986\,738\,826\,660\,532\,9

In

\mathrm{Simplify}(\operatorname{Normal}(\operatorname{Series}(\mathrm{Hypergeometric1F1Regularized}(1/2, 3, x), x, ∞, 3)))

Out

\operatorname{Series}(\mathrm{Hypergeometric1F1Regularized}(\frac{1}{2}, 3, x), x, ∞, 3)

Scope #

In

N(\mathrm{Hypergeometric1F1Regularized}(2, -4, 2), 50)

Out

1\,891.598\,361\,326\,246\,458\,170\,989\,429\,907\,202\,000\,174\,160\,786\,061\,3

In

N(\mathrm{Hypergeometric1F1Regularized}(1, 0, 1), 30)

Out

2.718\,281\,828\,459\,045\,235\,360\,287\,471\,35

In

\mathrm{MatrixFunction}(()↦\mathrm{Hypergeometric1F1Regularized}(\frac{1}{2}, 1, \operatorname{\_1}), \bigl\lbrack\bigl\lbrack\frac{1}{2}, 0\bigr\rbrack, \bigl\lbrack0, \frac{1}{2}\bigr\rbrack\bigr\rbrack)

Out

\bigl\lbrack\bigl\lbrack\operatorname{I}_{0}(\frac{1}{4})\sqrt[4]{e}, 0\bigr\rbrack, \bigl\lbrack0, \operatorname{I}_{0}(\frac{1}{4})\sqrt[4]{e}\bigr\rbrack\bigr\rbrack

Properties #

In

N(\frac{1}{2}(\mathrm{Hypergeometric1F1Regularized}(3/2, 1/2, 1/2))-\mathrm{Hypergeometric1F1Regularized}(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}))

Out

0.0

Known value · Wolfram MathematicalFunctionData, Hypergeometric1F1Regularized, FunctionalEquations

In

N(-\mathrm{Hypergeometric1F1Regularized}(\frac{1}{2}, -\frac{1}{2}, \frac{1}{2}))

Out

0.0

Known value · Wolfram MathematicalFunctionData, Hypergeometric1F1Regularized, FunctionalEquations

In

N(-\frac{1}{2}(\mathrm{Hypergeometric1F1Regularized}(\frac{1}{2}, \frac{3}{2}, \frac{1}{2}))-\mathrm{Hypergeometric1F1Regularized}(-\frac{1}{2}, \frac{1}{2}, \frac{1}{2})+\mathrm{Hypergeometric1F1Regularized}(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}))

Out

-1.0\cdot10^{-16}

Known value · Wolfram MathematicalFunctionData, Hypergeometric1F1Regularized, FunctionalEquations

In

N(-\frac{1}{2}(\mathrm{Hypergeometric1F1Regularized}(\frac{3}{2}, \frac{1}{2}, \frac{1}{2}))+\mathrm{Hypergeometric1F1Regularized}(\frac{1}{2}, -\frac{1}{2}, \frac{1}{2})+\mathrm{Hypergeometric1F1Regularized}(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}))

Out

0.0

Known value · Wolfram MathematicalFunctionData, Hypergeometric1F1Regularized, FunctionalEquations

About

Examples	16
Checked against	Wolfram Language 15 agree1 unevaluated
Crosswalk	Fungrim Fungrim entry ×2 Wolfram Language

Implementation

primitive · numericirreducibly numeric — it evaluates rather than rewrites

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

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

Hypergeometric1F1Regularized[a,b,z]; mpmath's own regularized series (rgamma per term).

See also: Hypergeometric2F1Regularized, HypergeometricU