BesselY

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

Bessel function of the second kind (Neumann function)

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaBessel function↗Bessel function of the second kind↗MathWorldBesselFunctionoftheSecondKind↗WikidataQ109545924↗DLMF10.2.3↗10.2.E3↗FungrimBesselY↗Fungrim entryWolfram LanguageBesselY✓↗

BesselY(order: complex, complex | infinity) -> numberAs native, and keeps its digits past a double's for integer order and positive real argument (N(BesselY(0, 1), 50)).

SageMathbessel_Y(order: complex, complex | infinity)mpmathbessely(order: complex, complex | infinity)Juliaenumeratio_ball(bessel_y, order: complex, complex | infinity)

Typed signatures

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

Domain: Compute engine

Plot

the complex plot cannot lower BesselY

Examples

Basic #

live ↗

In$N(\operatorname{Y}_{0}(1))$Out$0.088\,256\,964\,215\,677$

Known value · mpmath 1.3.0 bessely

live ↗

In$\operatorname{Normal}(\operatorname{Series}(\operatorname{Y}_{0}(x), x, ∞, 2))$Out$(\frac{-\frac{\sqrt{2}}{2}i}{\sqrt{π}\sqrt{x}}-\frac{\sqrt{2}}{16\sqrt{π}\sqrt{x}^{3}})e^{i(x-\frac{π}{4})}+(\frac{\frac{\sqrt{2}}{2}i}{\sqrt{π}\sqrt{x}}-\frac{\sqrt{2}}{16\sqrt{π}\sqrt{x}^{3}})e^{-i(x-\frac{π}{4})}$

Scope #

live ↗

In$N(\operatorname{Y}_{0}(1), 50)$Out$0.088\,256\,964\,215\,676\,957\,982\,926\,766\,023\,515\,162\,827\,817\,523\,090\,676$

live ↗

In$\bigl\lbrack\operatorname{Y}_{0}(0), \operatorname{Y}_{1}(0)\bigr\rbrack$Out$\bigl\lbrack-∞, \tilde∞\bigr\rbrack$

live ↗

In$N(\mathrm{FunctionRange}(\operatorname{Y}_{0}(x), x, y))$Out$y≤0.520\,786\,412\,402\,267\,5$

live ↗

In$N(\mathrm{FunctionRange}(\operatorname{Y}_{1}(x), x, y))$Out$y≤0.416\,729\,928\,106\,451\,4$

live ↗

In$\frac{\mathrm{d}\operatorname{Y}_{n}(x)}{\mathrm{d}x}$Out$\frac{\operatorname{Y}_{n-1}(x)}{2}-\frac{\operatorname{Y}_{n+1}(x)}{2}$

live ↗

In$\operatorname{Series}(\operatorname{Y}_{0}(x), x, 1, 3)$Out$\operatorname{Y}_{0}(1)+(x-1)(\frac{\operatorname{Y}_{-1}(1)}{2}-\frac{\operatorname{Y}_{1}(1)}{2})+\frac{1}{8}(-2\operatorname{Y}_{0}(1)+\operatorname{Y}_{2}(1)+\operatorname{Y}_{-2}(1))(x-1)^2+\frac{1}{48}(-3\operatorname{Y}_{-1}(1)+3\operatorname{Y}_{1}(1)-\operatorname{Y}_{3}(1)+\operatorname{Y}_{-3}(1))(x-1)^3+O\left((x-1)^4\right)$

About

Examples	8
Checked against	Wolfram Language 8 agreeSageMath 5 agree1 unevaluatedmpmath 2 agree1 differJulia 2 agree1 error
Crosswalk	Wikipedia ×2MathWorldWikidataDLMF ×2FungrimFungrim entry ×13Wolfram LanguageSageMathmpmathJulia

Implementation

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts