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Research draft

special function

vr.tr.special-function · XCT.QLT

Let an agent explain special functions as a category, identify major families and their origins, relay reference sources and computational methods, and distinguish special functions from elementary functions.

Thing Registry Cross-cutting context

Research draft, second pass

A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.

written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify

Researched by: Claude

Purpose and description

Let an agent explain special functions as a category, identify major families and their origins, relay reference sources and computational methods, and distinguish special functions from elementary functions.

A mathematical function with an established name and notation because of its importance in analysis, physics and applied mathematics, beyond the elementary functions, such as the gamma and beta functions, Bessel and modified Bessel functions, spherical Hankel functions, Legendre and associated Legendre polynomials, elliptic integrals, the error function, trigonometric integrals, Airy and Scorer functions and hypergeometric functions; special functions arise as solutions of differential equations and integrals and are tabulated in standard references.

What it is for: Solving equations and integrals in science and engineering.

It can be explain the category; identify major families; relay references and computation; distinguish from elementary functions.

Distinguishing features

Named and tabulated

Differential equation origins

Physical applications

Extensive identities

What it looks like

Not a visible object; named functions with graphs and tables.

How it is recognised

Named non-elementary functions

Gamma, Bessel, Legendre, elliptic, error, Airy, hypergeometric

Elementary functions are built from polynomials, exponentials and trigonometry

Related models

is a kind of - in registry terms

function

is contrasted with - the simpler class

elementary function

arises from - in physics

differential equation

is catalogued in - the standard reference

NIST Digital Library of Mathematical Functions

In practice

Families and kinds

gamma, beta and related functions

Bessel, modified Bessel and Hankel functions including spherical forms

orthogonal polynomials such as Legendre and associated Legendre

elliptic integrals and functions

error function and trigonometric integrals

Airy and Scorer functions

hypergeometric functions

zeta and related functions

Identifiers

DLMF chapters NIST reference

Standards and regulation

No regulation; notation conventions from the DLMF and ISO 80000-2

Failure modes and hazards

Notation and normalisation inconsistencies

Numerical instability in computation

Confusing families

Also called

spherical Hankel functionassociated Legendre polynomialsmodified Bessel functionScorer's functiontrigonometric integralelliptic integraltheta functionlog sine integralLegendre functionWigner 3-j symbolHankel functionWeierstrass functionsWhittaker functionAiry functionChebyshev polynomialspin-weighted spherical harmonicsspinor spherical harmonicsvector spherical harmonicsClebsch–Gordan coefficient

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Understand What special functions are.

Mathematics.

Definition

Definition and scope.

Definition

Definition.

  1. What makes a function special, and how does the category relate to elementary functions? definition
  2. Is the question about special functions in general or a specific family? boundary

Families

Major families.

Families

Families.

  1. What are the major families, and where does each arise? definition
  2. Which entry fits the specific function? action
Use Applications.

Application.

Physics

Physics and engineering.

Physics

Physics.

  1. How do Bessel, Legendre, Airy and hypergeometric functions arise in physics and engineering problems? provenance
  2. Which entry fits the specific application? action

Statistics

Statistics and probability.

Statistics

Statistics.

  1. How do the gamma, beta and error functions appear in statistics? provenance
  2. Which references are standard? provenance
Compute Computation and references.

Practice.

Computation

Numerical computation.

Computation

Computation.

  1. How are special functions computed reliably in software libraries? provenance
  2. Which sources are cited? provenance

References

Reference works.

References

References.

  1. What are the standard references, from Abramowitz and Stegun to the DLMF? provenance
  2. Which entry fits the DLMF? action
Context History and theory.

Context.

History

History.

History

History.

  1. How did special functions develop from Euler and Bessel to the twentieth century? provenance
  2. Which entry fits the history of analysis? action

Theory

Unifying theory.

Theory

Theory.

  1. How do hypergeometric functions and group theory unify many special functions? provenance
  2. Which entry fits hypergeometric function? action

What the second pass must settle

  • Should each family be a separate entry?
  • How should the DLMF be linked?
  • The registry entry has merged aliases naming specific functions; should they be split off?
Draft as JSON