Comparison of Special Functions

Introduction

This article compares the special functions of the eight algebras $\mathbb{R}, \mathbb{C}, \mathbb{D}, \mathbb{D}', \mathbb{H}, \mathbb{H}_{\mathrm{s}}, \mathbb{B}, \mathbb{H}_{\mathbb{D}}$: the exponential, the logarithm, the trigonometric and hyperbolic functions and their hypercomplex analogues; the Gamma, Beta, error and zeta functions of a hypercomplex argument; and the orthogonal polynomials, the Bessel, Airy and hypergeometric functions and the elliptic integrals in each system. It states the situation of the eight algebras on each of these subjects in tables with the eight algebras as columns in the fixed order of The Eight Algebras Compared. Every entry restates a result of the special-function articles cited in the explanations.

The organising thread is the branching of the logarithm, and it is carried by the norm of each algebra. The exponential is never zero and is a local bijection; its kernel is the source of the multivaluedness of the logarithm, and the kernel is measured by the norm: an infinite cyclic group in $\mathbb{C}$, a family of spheres of roots of $-1$ in $\mathbb{H}$, a union of spheres and hyperboloids in $\mathbb{H}_{\mathrm{s}}$, an eigenvalue lattice in $\mathbb{B}$, and the pair of the quaternionic kernels in $\mathbb{H}_{\mathbb{D}}$. Where the algebra is split or degenerate, the trigonometric functions are replaced by the hyperbolic ones, the orthogonality of the polynomials fails because there is no definite form, and the whole theory decomposes along the idempotents or the fibres into copies of a lower system.

The Elementary Functions

The following table compares the elementary functions of the eight algebras: the exponential, the trigonometric and hyperbolic functions, the logarithm with its branching, and the powers. The eight algebras are the columns, in the fixed order.

datum	$\mathbb{R}$	$\mathbb{C}$	$\mathbb{D}$	$\mathbb{D}'$	$\mathbb{H}$	$\mathbb{H}_{\mathrm{s}}$	$\mathbb{B}$	$\mathbb{H}_{\mathbb{D}}$
exponential	$e^a$, entire	entire, $2\pi i$-periodic	factors $e^{a}e^{jb}$	truncates, $e^{a}(1+\varepsilon b)$	$e^{q_0}(\cos\lvert\mathbf v\rvert+\hat{\mathbf v}\sin\lvert\mathbf v\rvert)$	closed form in $c_0,c_1$ of $N(\mathbf v)$	values in $\mathbb{C}[\tilde Q]$, two regimes	$\exp(\tilde Q_+)\tilde\Pi_++\exp(\tilde Q_-)\tilde\Pi_-$
trigonometric, hyperbolic	classical	classical, from the exponential	hyperbolic replaces trigonometric	as real, truncated	parity parts of the exponential	parity parts, three branches	Pythagorean etc. for one argument	componentwise
logarithm	single-valued on $a>0$	multivalued, $2\pi i\mathbb{Z}$	two real logarithms	$\log a+\varepsilon b/a$	multivalued, $2\pi k$ on the roots of $-1$	on the image of $\exp$, multivalued at real scalars and $N(\operatorname{Vec})>0$	on the units	componentwise in the halves
powers	$a^b$	$A^a$	$A^a$	$\exp(a\log Z)$	$\exp(a\log\tilde q)$	inherit the ambiguity of the logarithm	$\tilde Q^\alpha$	$\tilde Q^\alpha$ componentwise

The table records the elementary theory. In $\mathbb{R}$ and $\mathbb{C}$ the exponential is entire and the trigonometric and hyperbolic series are built from it; the complex logarithm is the first multivalued function, its values differing by $2\pi i\mathbb{Z}$, and it is the model of the analytic continuation whose multivaluedness is measured by the periods of the exponential (Real Special Functions; Complex Special Functions). In $\mathbb{D}$ the exponential factors as $e^{a}e^{jb}$, so the trigonometric functions are replaced by their hyperbolic analogues because $j^2=+1$, and the logarithm is a pair of real logarithms in the idempotent coordinates; in $\mathbb{D}'$ the exponential sequence truncates at the first order and the logarithm is $\log a+\varepsilon b/a$, the infinitesimal part being the derivative of the real part times the infinitesimal coordinate (Split-Complex Special Functions; Dual-Numbers Special Functions). In $\mathbb{H}$ the exponential is $e^{q_0}(\cos\lvert\mathbf v\rvert+\hat{\mathbf v}\sin\lvert\mathbf v\rvert)$, its kernel is the family of spheres $2\pi k\Sigma$ of the roots of $-1$, and the logarithm is multivalued exactly by that kernel; the trigonometric and hyperbolic functions are the parity parts of the exponential, and the identities hold for a single argument because every element lies in a commutative plane (Quaternion Special Functions). In $\mathbb{H}_{\mathrm{s}}$ the exponential is never zero, its image is the set of elements of positive norm whose scalar part exceeds $-\sqrt{N}$ together with the negative scalars, so it is not surjective onto the units; the logarithm exists on that image, is a pair of real logarithms in the null coordinates, and is multivalued exactly at the non-zero real scalars and at the elements with $N(\operatorname{Vec}\tilde q)>0$; the addition formulas hold exactly for commuting arguments and otherwise fail by the Baker–Campbell–Hausdorff correction (Split-Quaternion Elementary Functions). In $\mathbb{B}$ every elementary function is the scalar power series applied to $\tilde Q$, so its values lie in the commutative subalgebra $\mathbb{C}[\tilde Q]$ and the standard identities survive for a single variable, while the addition formulas for two independent biquaternions fail; the subalgebra has two regimes, the oscillatory regime $B\neq0$ in which it is $\mathbb{C}\times\mathbb{C}$ with eigenvalues $Q_0\pm iB$ and the nilpotent regime $B=0$ in which every power series truncates after the linear term (Biquaternion Elementary Functions). In $\mathbb{H}_{\mathbb{D}}$ the elementary functions are computed in the idempotent form $\exp(\tilde Q_+)\tilde\Pi_++\exp(\tilde Q_-)\tilde\Pi_-$, which reduces them to the elementary functions of two ordinary quaternions; this is the primary form, the standard form in the scalar–vector decomposition being the secondary one (Split-Biquaternion Elementary Functions). The logarithm row records the branching: it is the kernel of the exponential in each column, infinite cyclic in $\mathbb{C}$, a lattice of eigenvalues in $\mathbb{B}$, and measured in every column by the norm.

The Higher Special Functions

The following table compares the higher special functions of the eight algebras: the Gamma and Beta functions, the zeta function, the error and Airy functions, the orthogonal polynomials, the Bessel, hypergeometric and Lambert functions, the elliptic integrals and the kernels from the analysis. The eight algebras are the columns, in the fixed order, with the marker — where the function is not carried.

datum	$\mathbb{R}$	$\mathbb{C}$	$\mathbb{D}$	$\mathbb{D}'$	$\mathbb{H}$	$\mathbb{H}_{\mathrm{s}}$	$\mathbb{B}$	$\mathbb{H}_{\mathbb{D}}$
Gamma, Beta	yes	yes	yes	yes	yes	integral on the commutative subalgebras, otherwise by its eigenvalues	integral, special cases	integral, special cases
zeta	yes	yes	—	—	—	—	special cases	special cases
error, Airy	yes	yes	yes	yes	yes	—	—	—
orthogonal polynomials	yes	yes	—	—	—	substitution, orthogonality fails	series	componentwise
Bessel	yes	yes	yes	yes	yes	series	series, closed in the two regimes	componentwise
hypergeometric	yes	yes	yes	yes	yes	series	series	componentwise
elliptic integrals	yes	yes	—	—	—	—	—	—
Lambert $W$	yes	yes	yes	yes	yes	—	—	—
kernels from the analysis	—	—	—	—	monogenic functions	Poisson and Legendre, characters	Cauchy, Poisson, Green kernels	componentwise kernels

The table shows how far the higher theory extends. In $\mathbb{R}$ and $\mathbb{C}$ the whole catalogue is present: the integral representations of the Gamma and Beta functions, the Dirichlet series of the zeta function, the error and Airy functions, the orthogonal polynomials, the Bessel and hypergeometric functions with their differential equations, the elliptic integrals and the Lambert $W$ function as the multivalued inverse of $w\mapsto we^w$ (Real Special Functions; Complex Special Functions). In $\mathbb{D}$ and $\mathbb{D}'$ the functions are carried by the substitution principle: the hyperbolic or truncated forms of the Gamma, Beta, error, Airy, Bessel and hypergeometric functions exist, and every dual special function is the ordinary real function of the real part plus its derivative times the infinitesimal part, but the zeta function, the orthogonal polynomials and the elliptic integrals are not carried (Split-Complex Special Functions; Dual-Numbers Special Functions). In $\mathbb{H}$ the Gamma, Beta, error, Airy, Bessel, hypergeometric and Lambert $W$ functions exist with their differential equations and the monogenic special functions of the analysis, but the zeta function, the orthogonal polynomials and the elliptic integrals are not carried (Quaternion Special Functions). In $\mathbb{H}_{\mathrm{s}}$ the power-series functions extend by substitution and reduce on the commutative subalgebras, the Gamma function is defined by its integral on the subalgebras and elsewhere by the interpolation on its eigenvalues, and the functions from the analysis — the fundamental-solution kernel, the Poisson kernel, the Legendre functions and the characters of the norm-one group — complete the catalogue; the orthogonality of the polynomials fails, because the algebra has no definite form and no positive measure (Split-Quaternion Higher Special Functions). In $\mathbb{B}$ and $\mathbb{H}_{\mathbb{D}}$ the higher special functions fall into three classes: the power-series functions (Bessel, orthogonal polynomials, hypergeometric) are summed in closed form in the two pure regimes, the integral-defined functions (Gamma, zeta) are established only in special cases and their functional equations are open, and the functions from the analysis (Cauchy, Poisson and Green kernels) are standard but otherwise open; in $\mathbb{B}$ the common thread is the complex norm $\theta$ of the vector part, and in $\mathbb{H}_{\mathbb{D}}$ it is the idempotent decomposition, which reduces the power-series functions to the corresponding functions of two ordinary quaternions and makes the theory simpler than the biquaternion case (Biquaternion Higher Special Functions; Split-Biquaternion Higher Special Functions). The elliptic-integral row is carried by $\mathbb{R}$ and $\mathbb{C}$ alone, and the zeta row by the same two columns plus the two biquaternion systems in special cases.

Summary

The exponential is entire and never zero in every column, and its kernel is the source of the multivaluedness of the logarithm: infinite cyclic in $\mathbb{C}$, a family of spheres of roots of $-1$ in $\mathbb{H}$, a union of spheres and hyperboloids in $\mathbb{H}_{\mathrm{s}}$, an eigenvalue lattice in $\mathbb{B}$, and the pair of the quaternionic kernels in $\mathbb{H}_{\mathbb{D}}$; in $\mathbb{D}$ and $\mathbb{D}'$ the theory decomposes along the idempotents or the fibres into copies of the real line. The trigonometric functions are replaced by the hyperbolic ones where the square of the unit is $+1$ or $0$, the identities hold for a single argument in a commutative plane or subalgebra and fail for two independent arguments, and the addition formulas fail exactly by the Baker–Campbell–Hausdorff correction. The Gamma, Beta, Bessel and hypergeometric functions are carried by all eight columns; the error and Airy functions and the Lambert $W$ function are carried by the five columns $\mathbb{R}$, $\mathbb{C}$, $\mathbb{D}$, $\mathbb{D}'$ and $\mathbb{H}$ and are not carried by the three split higher systems; the zeta function and the elliptic integrals are carried by $\mathbb{R}$ and $\mathbb{C}$ alone, with the zeta function appearing in special cases in the two biquaternion systems; and the orthogonal polynomials lose their orthogonality in $\mathbb{H}_{\mathrm{s}}$, where there is no definite form, though they survive by substitution. The functions from the analysis — the monogenic functions in $\mathbb{H}$, the Poisson and Legendre functions in $\mathbb{H}_{\mathrm{s}}$, the Cauchy, Poisson and Green kernels in $\mathbb{B}$ and $\mathbb{H}_{\mathbb{D}}$ — complete the catalogue in the four higher systems, and are the only special functions of those systems that are intrinsically hypercomplex rather than the transport of a real or complex definition.

Summary of Notation