notatio

BesselJ ​

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

Bessel function of the first kind

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaBessel functionMathWorldBesselFunctionoftheFirstKindBesselFunctionWikidataQ219637nLabBessel functionBritannicatopic/Bessel-functionDLMF10.2.E2FungrimBesselJ✓Fungrim entryWolfram LanguageBesselJ✓
BesselJ(order: complex, complex | infinity) -> numberAs native, and keeps its digits past a double's for integer order and real argument (N(BesselJ(1, 300), 25)).
SageMathbessel_J(order: complex, complex | infinity)mpmathbesselj(order: complex, complex | infinity)Juliaenumeratio_ball(bessel_j, order: complex, complex | infinity)
Typed signatures
TypePackageApplies
(order: complex, complex | infinity) -> numberanalyticreplaces compute-enginealways
(order: complex, complex | infinity) -> numbercompute-enginealways

Domain: Compute engine

Plot

ComplexPlot
real slice

Examples

Basic #
live ↗
InN(J⁡0(1))N(\operatorname{J}_{0}(1))Out0.765 197 686 557 966 60.765\,197\,686\,557\,966\,6

Known value · mpmath 1.3.0 besselj

live ↗
InNormal⁡(Series⁡(J⁡0(x),x,∞,3))\operatorname{Normal}(\operatorname{Series}(\operatorname{J}_{0}(x), x, ∞, 3))Out(2πx−92128πx5)cos⁡(π4−x)−2sin⁡(π4−x)8πx3(\frac{\sqrt{2}}{\sqrt{π}\sqrt{x}}-\frac{9\sqrt{2}}{128\sqrt{π}\sqrt{x}^{5}})\cos(\frac{π}{4}-x)-\frac{\sqrt{2}\sin(\frac{π}{4}-x)}{8\sqrt{π}\sqrt{x}^{3}}
Scope #
live ↗
InN(J⁡0(4),50)N(\operatorname{J}_{0}(4), 50)Out−0.397 149 809 863 847 372 286 590 768 451 698 041 975 618 685 289 39-0.397\,149\,809\,863\,847\,372\,286\,590\,768\,451\,698\,041\,975\,618\,685\,289\,39
live ↗
IndJ⁡n(x)dx\frac{\mathrm{d}\operatorname{J}_{n}(x)}{\mathrm{d}x}OutJ⁡n−1(x)2−J⁡n+1(x)2\frac{\operatorname{J}_{n-1}(x)}{2}-\frac{\operatorname{J}_{n+1}(x)}{2}
About
Examples4
Checked againstWolfram Language 4 agreeSageMath 3 agree1 unevaluatedmpmath 2 agreeJulia 2 agree
CrosswalkWikipediaMathWorld ×2WikidatanLabBritannicaDLMFFungrimFungrim entry ×26Wolfram LanguageSageMathmpmathJulia

Implementation

mappedmpmathpackages/oracle/src/mappings.ts
mappedsagepackages/oracle/src/mappings.ts
mappedjuliapackages/oracle/src/mappings.ts