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

SymPyhyper([a], [b], z) / gamma(b)mpmathenumeratio_hyperreg([a], [b], z)Juliaenumeratio_ball(hypergeometric_1f1_regularized, a, b, z)

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 $d$.

Plot

Examples

9 more kept as data, checked against the oracles

Basic #

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

live ↗

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

live ↗

In$\mathrm{Hypergeometric1F1Regularized}(1, -1, 0.7)$Out$0.986\,738\,826\,660\,532\,9$

live ↗

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 #

live ↗

In$N(\mathrm{Hypergeometric1F1Regularized}(2, -4, 2), 50)$Out$1\,891.598\,361\,326\,246\,458\,170\,989\,429\,907\,202\,000\,174\,160\,786\,061\,3$

live ↗

In$N(\mathrm{Hypergeometric1F1Regularized}(1, 0, 1), 30)$Out$2.718\,281\,828\,459\,045\,235\,360\,287\,471\,35$

live ↗

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

live ↗

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

live ↗

In$N(-\mathrm{Hypergeometric1F1Regularized}(\frac{1}{2}, -\frac{1}{2}, \frac{1}{2}))$Out$0.0$

Known value · Wolfram MathematicalFunctionData, Hypergeometric1F1Regularized, FunctionalEquations

live ↗

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$0.0$

Known value · Wolfram MathematicalFunctionData, Hypergeometric1F1Regularized, FunctionalEquations

live ↗

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	19
Checked against	Wolfram Language 16 agree1 unevaluatedSymPy 6 agree7 differ3 unevaluated2 errormpmath 15 agree2 errorJulia 17 agree
Crosswalk	FungrimFungrim entry ×2Wolfram LanguageSymPympmathJulia

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).

mappedmpmathpackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

SymPy has no regularized hypergeometric, so this divides by Γ(b); at a pole of Γ(b) (b a non-positive integer) the quotient is not the limit that ours is.

See also: Hypergeometric2F1Regularized, HypergeometricU