BesselK

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

Modified Bessel function of the second kind (Macdonald function)

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaModified Bessel function of the second kind↗MathWorldModifiedBesselFunctionoftheSecondKind↗WikidataQ109559130↗DLMF10.25.E3↗FungrimBesselK↗Fungrim entryWolfram LanguageBesselK✓↗

BesselK(order: complex, complex | infinity) -> numberAs native, and keeps its digits past a double's for integer order and positive real argument (N(BesselK(0, 4), 50)).

SageMathbessel_K(order: complex, complex | infinity)mpmathbesselk(order: complex, complex | infinity)Juliaenumeratio_ball(bessel_k, 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 BesselK

Examples

Basic #

live ↗

In$N(\operatorname{K}_{0}(1))$Out$0.421\,024\,438\,240\,708\,34$

Known value · mpmath 1.3.0 besselk

live ↗

In$\operatorname{Normal}(\operatorname{Series}(\operatorname{K}_{0}(x), x, ∞, 4))$Out$(-\frac{\sqrt{2}\sqrt{π}}{16\sqrt{x}^{3}}-\frac{75\sqrt{2}\sqrt{π}}{2\,048\sqrt{x}^{7}}+\frac{9\sqrt{2}\sqrt{π}}{256\sqrt{x}^{5}}+\frac{\sqrt{2}\sqrt{π}}{2\sqrt{x}})e^{-x}$

Scope #

live ↗

In$N(\operatorname{K}_{0}(4), 50)$Out$0.011\,159\,676\,085\,853\,024\,269\,745\,195\,979\,833\,489\,225\,009\,023\,888\,474$

live ↗

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

live ↗

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

live ↗

In$\operatorname{Normal}(\operatorname{Series}(\operatorname{K}_{n}(x), x, ∞, 2))$Out$(\frac{\sqrt{2}(4n^2-1)\sqrt{π}}{16\sqrt{x}^{3}}+\frac{\sqrt{2}\sqrt{π}}{2\sqrt{x}})e^{-x}$

About

Examples	6
Checked against	Wolfram Language 6 agreeSageMath 4 agree2 unevaluatedmpmath 2 agree1 differJulia 2 agree1 error
Crosswalk	WikipediaMathWorldWikidataDLMFFungrimFungrim entry ×14Wolfram LanguageSageMathmpmathJulia

Implementation

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts