Hypergeometric1F1

Generated from the engine's own definition: compute-engine's symbol, which we neither extend nor document by hand.

Kummer confluent hypergeometric function ₁F₁(a; b; z) = M(a, b, z).

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaConfluent hypergeometric function↗MathWorldConfluentHypergeometricFunctionoftheFirstKind↗WikidataQ783948↗Encyclopedia of MathematicsConfluent_hypergeometric_equation↗FungrimHypergeometric1F1✓↗Fungrim entry2df3e3↗6cf802↗98688d✓↗9ad254↗abadc7✓↗be533c↗Wolfram LanguageHypergeometric1F1✓↗

Hypergeometric1F1(complex | infinity, complex | infinity, complex | infinity) -> numberKummer confluent hypergeometric function ₁F₁(a; b; z) = M(a, b, z).

SageMathhypergeometric([complex | infinity], [complex | infinity], complex | infinity)mpmathhyp1f1(complex | infinity, complex | infinity, complex | infinity)Juliaenumeratio_ball(hypergeometric_1f1, complex | infinity, complex | infinity, complex | infinity)

Typed signatures

Type	Package	Applies
(complex | infinity, complex | infinity, complex | infinity) -> number	analyticreplaces compute-engine	always
(complex | infinity, complex | infinity, complex | infinity) -> number	compute-engine	always

Domain: Compute engine

Details

Kummer's confluent hypergeometric function $M(a,b,z) = {}_1F_1(a;b;z) = \sum_{k\ge0} (a)_k/(b)_k \cdot z^k/k!$ (DLMF 13.2.2) is entire in $z$ for every $a, b$ — the confluence of Hypergeometric2F1's regular singular points at $1$ and $\infty$ into one irregular singular point at $\infty$, taking $z \to z/b$ and $b \to \infty$ in ${}_2F_1(a,b;c;z/b)$. It solves Kummer's equation $z w'' + (b-z) w' - a w = 0$; $U(a,b,z)$ (see HypergeometricU) is the equation's other standard solution. It is compute-engine's own native head, declared for real and complex $z$.

• $M(a,b,0) = 1$ for every $a, b$.
• Kummer's transformation: $M(a,b,z) = e^z\,M(b-a,b,-z)$ (DLMF 13.2.39).
• $M(1,2,2z) = e^z\sinh(z)/z$ (DLMF 13.6.2), so $M(1,2,z) = (e^z-1)/z$.
• $\operatorname{erf}(z) = \dfrac{2z}{\sqrt\pi}\,M(\tfrac12,\tfrac32,-z^2)$ (DLMF 7.11.4); see Erf.
• $a$ a nonpositive integer terminates the series into a polynomial in $z$ of that degree, any $b$ not itself a nonpositive integer at or before that degree (where the denominator would divide by zero first).

Plot

ComplexPlot

real slice

Examples

Basic #

The series is $1$ at the origin

live ↗

In$\mathrm{Hypergeometric1F1}(1, 1, 0)$Out$1$

${}_1F_1(1; 2; 1) = e - 1$

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In$N(\mathrm{Hypergeometric1F1}(1, 2, 1), 20)$Out$1.718\,281\,828\,459\,045\,235\,4$

Known value · mpmath 1.3.0 hyp1f1

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In$\operatorname{Series}(\mathrm{Hypergeometric1F1}(a, b, x), x, 0, 3)$Out$1+x\mathrm{Hypergeometric1F1}^{(0, 0, 1)}(a, b, 0)+\frac{1}{2}\mathrm{Hypergeometric1F1}^{(0, 0, 2)}(a, b, 0)x^2+\frac{1}{6}\mathrm{Hypergeometric1F1}^{(0, 0, 3)}(a, b, 0)x^3+O\left(x^4\right)$

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In$\mathrm{FullSimplify}(\operatorname{Normal}(\operatorname{Series}(\mathrm{Hypergeometric1F1}(a, b, x), x, ∞, 2)))$Out$\frac{(-a+b+1)(1-a)(2-a)(b-a)Γ(b)e^{x}x^{a-b}}{2Γ(a)x^2}+\frac{\frac{a×(a-b+1)(a-b+2)(a+1)Γ(b)}{(-x)^{a}}}{2Γ(b-a)x^2}+\frac{(1-a)(b-a)Γ(b)e^{x}x^{a-b}}{xΓ(a)}-\frac{\frac{a×(a-b+1)Γ(b)}{(-x)^{a}}}{xΓ(b-a)}+\frac{1}{Γ(a)}Γ(b)e^{x}x^{a-b}+\frac{\frac{Γ(b)}{(-x)^{a}}}{Γ(b-a)}$

Scope #

live ↗

In$N(\mathrm{Hypergeometric1F1}(10, \frac{1}{3}, -1), 50)$Out$1.085\,646\,966\,277\,114\,418\,106\,099\,920\,005\,389\,482\,134\,181\,965\,565\,5$

live ↗

In$\mathrm{Hypergeometric1F1}(1, 1, \bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack0, 1\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack e, 1\bigr\rbrack, \bigl\lbrack1, e\bigr\rbrack\bigr\rbrack$

live ↗

In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{Hypergeometric1F1}(1, 1, \operatorname{\_1}), \bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack0, 1\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack e, 0\bigr\rbrack, \bigl\lbrack0, e\bigr\rbrack\bigr\rbrack$

About

Examples	7
Checked against	Wolfram Language 7 agreeSageMath 5 agree1 unevaluatedmpmath 4 agreeJulia 4 agree
Crosswalk	WikipediaMathWorldWikidataEncyclopedia of MathematicsFungrimFungrim entry ×6Wolfram LanguageSageMathmpmathJulia

Implementation

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts