Octonion Special Functions
Introduction
This article is the special functions slot of the octonion system. It sets out the elementary and higher functions of an octonion variable — the exponential, the trigonometric and hyperbolic functions, the logarithm and powers, the gamma function and its relatives, and the class of functions that survives the failure of associativity, namely the axial and the monogenic functions constructed from holomorphic ones.
The article is the octonion member of the special functions slots of this Part and follows the model of Quaternion Special Functions, Split-Biquaternion Elementary Functions and Split-Biquaternion Higher Special Functions. It takes the operator $D$, the monogenic class and the kernel from Octonion Analysis and the constructions of Octonion Integration; the harmonic analysis of these functions is the subject, which closes the octonion ladder.
Conventions. $\mathbb{O}$ with basis $e_0,\dots,e_7$, $e_k^2 = -e_0$ for $k\geq1$; $\tilde o = o_0 + \mathbf{o}$ with $o_0 = \operatorname{Sc}\tilde o\in\mathbb{R}$ and $\mathbf{o}\in\operatorname{Im}\mathbb{O}$; conjugation $\tilde o^{\natural}$, norm $\lvert \tilde o\rvert$, inner product $\langle \tilde o,\tilde p\rangle = \operatorname{Sc}(\tilde o \tilde p^{\natural})$; the powers of a single element are the unambiguous left-associated powers, $\tilde o^n = \tilde o(\tilde o(\cdots \tilde o))$, which coincide with the right-associated powers by power-associativity. The operator $D = \sum_{k=0}^{7}e_k\partial_k$ is the Cauchy–Riemann operator of Octonion Analysis. Throughout, "series in $\tilde o$" means a series whose general term is a real multiple of a power $\tilde o^n$, so that no bracketing question arises; series in two or more octonion variables are marked where they occur.
Power Series and Elementary Functions
Convergence and Power-Associativity
Definition. Let $\sum a_nX^n$ be a real power series with radius of convergence $R$ (possibly infinite). The associated octonionic function is
$$ f(\tilde o) = \sum_{n=0}^{\infty}a_n\tilde o^n , \qquad \lvert \tilde o\rvert < R , $$
with $\tilde o^n$ the unambiguous power of the single element $\tilde o$.
Lemma. The powers $\tilde o^n$ are well defined for every $n$ and satisfy $\tilde o^m\tilde o^n = \tilde o^{m+n}$ for all $m,n\geq0$; the subalgebra generated by $\tilde o$ and $e_0$, that is $\mathbb{R}[\tilde o] = \{a + b\tilde o\}$, is commutative and associative; hence all algebraic manipulations of the terms of a one-variable power series are legitimate.
Proof. The octonions are power-associative: the subalgebra generated by a single element and the identity is associative, and it is commutative because it is spanned by $e_0$ and $\tilde o$, whose products $\tilde o^2 = (o_0+\mathbf{o})^2 = o_0^2 + 2o_0\mathbf{o} - \lvert\mathbf{o}\rvert^2$ and $\tilde o^3 = \tilde o^2\tilde o$ lie in the span; the standard proof that a one-generator algebra with a bilinear product is associative in the alternative case is Artin's theorem, as in Octonion Algebra.
Proposition. For the series above: the function $f$ is real-analytic, its norm satisfies $\lvert f(\tilde o)\rvert\leq\sum_n\lvert a_n\rvert\lvert \tilde o\rvert^n$, the convergence is absolute and uniform on compact subsets of the ball $\lvert \tilde o\rvert Proof. The absolute convergence follows from the submultiplicativity $\lvert \tilde o\tilde p\rvert = \lvert \tilde o\rvert\lvert \tilde p\rvert$ of the norm, which holds for the octonions as for any composition algebra and was established in Octonion Norm and Invertibility; the analyticity of each term in the eight real coordinates follows from that of the powers; the identification of the coefficients is the restriction to the real axis times $e_0$. Definition. The exponential, sine, cosine, hyperbolic sine and hyperbolic cosine of $\tilde o$ are $$
\exp \tilde o = \sum_{n=0}^{\infty}\frac{\tilde o^n}{n!}, \qquad
\sin \tilde o = \sum_{k=0}^{\infty}\frac{(-1)^k\tilde o^{2k+1}}{(2k+1)!}, \qquad
\cos \tilde o = \sum_{k=0}^{\infty}\frac{(-1)^k\tilde o^{2k}}{(2k)!},
$$
$$
\sinh \tilde o = \sum_{k=0}^{\infty}\frac{\tilde o^{2k+1}}{(2k+1)!}, \qquad
\cosh \tilde o = \sum_{k=0}^{\infty}\frac{\tilde o^{2k}}{(2k)!},
$$ each convergent for all $\tilde o\in\mathbb{O}$, together with the tangent and the remaining functions formed from them by the usual quotients. Theorem. The following identities hold for all $\tilde o\in\mathbb{O}$. Proof. Every identity is an identity between series whose terms are powers of the single element $\tilde o$; all terms lie in the commutative associative algebra $\mathbb{R}[\tilde o]$ by the lemma, and in that algebra the proofs are the classical ones, term by term in the absolute convergence. For (4), $\tilde o^{2k} = (-1)^k\lvert \tilde o\rvert^{2k}e_0$ and $\tilde o^{2k+1} = (-1)^k\lvert \tilde o\rvert^{2k}\tilde o$, so the exponential series becomes $\sum_k(-1)^k\lvert \tilde o\rvert^{2k}/(2k)! + \frac{\tilde o}{\lvert \tilde o\rvert}\sum_k(-1)^k\lvert \tilde o\rvert^{2k+1}/(2k+1)!$, which is the display; the trigonometric series give the hyperbolic values by the same substitution. The fourth item deserves a remark, because it is the exact point at which the non-commutativity of the octonions, rather than their non-associativity, becomes visible in the elementary functions. The identity $\exp \tilde o = \cos \tilde o + \sin \tilde o$ of item 3 expresses the exponential of a general octonion through the trigonometric functions of that octonion; the identity of item 4 expresses the exponential of an imaginary octonion through the real trigonometric functions of the real number $\lvert \tilde o\rvert$. The two displays are consistent — the passage from the first to the second uses $\cos \tilde o = \cosh\lvert \tilde o\rvert$ and $\sin \tilde o = \frac{\tilde o}{\lvert \tilde o\rvert}\sinh\lvert \tilde o\rvert$, so that $\cos \tilde o + \sin \tilde o = \cosh\lvert \tilde o\rvert + \frac{\tilde o}{\lvert \tilde o\rvert}\sinh\lvert \tilde o\rvert$, which is a different-looking but equal expression — and they are the octonionic form of the classical complex identities $e^{i\theta} = \cos\theta + i\sin\theta$ and $\cos(ia) = \cosh a$, $\sin(ia) = i\sinh a$. The direction $\tilde o/\lvert \tilde o\rvert$ plays the rôle of the imaginary unit, and every direction does so equally, which is the content of the rotational symmetry of the octonion norm. Definition. For $\tilde o$ with $\lvert \tilde o\rvert > 0$ a logarithm of $\tilde o$ is an octonion $u$ with $\exp u = \tilde o$; it is written $\log \tilde o$ when the set of logarithms is meant, and the principal logarithm is the logarithm $u = \ln\lvert \tilde o\rvert + \theta\hat{\mathbf{o}}$ with $\theta\in[0,\pi]$. The powers are $\tilde o^{\alpha} = \exp(\alpha\log \tilde o)$ for real $\alpha$ and the principal branch. Theorem. Every non-zero octonion has a logarithm, and the set of logarithms of $\tilde o$ is as follows: if $\operatorname{Vect}(\tilde o)\neq0$, there is exactly one logarithm, namely $\ln\lvert \tilde o\rvert + \arccos(o_0/\lvert \tilde o\rvert)\hat{\mathbf{o}}$; if $\tilde o$ is a positive real multiple of $e_0$, the logarithms are $\ln\lvert \tilde o\rvert + u$ with $u$ any imaginary octonion of norm $2\pi$; if $\tilde o$ is a negative real multiple of $e_0$, the logarithms are $\ln\lvert \tilde o\rvert + u$ with $u$ any imaginary octonion of norm $\pi$. In each case $\exp(\log \tilde o) = \tilde o$, the powers satisfy $\tilde o^{\alpha}\tilde o^{\beta} = \tilde o^{\alpha+\beta}$ with unambiguous bracketing, and the principal logarithm satisfies $\log(\exp \tilde o) = \tilde o$ for $\lvert \tilde o\rvert<\pi$. Proof. Write $c = \tilde o/\lvert \tilde o\rvert$ with $\lvert c\rvert = 1$ and $c = c_0 + \underline c$; by item (4) of the previous theorem, $\exp u = \cos\lvert u\rvert + \frac{u}{\lvert u\rvert}\sin\lvert u\rvert$, so $\exp u = c$ forces $\cos\lvert u\rvert = c_0$ and $\frac{u}{\lvert u\rvert}\sin\lvert u\rvert = \underline c$. When $\underline c\neq0$ the norm $\lvert u\rvert = \arccos c_0$ is determined and the direction $\hat u = \underline c/\lvert\underline c\rvert$ is determined; when $\underline c = 0$ one has $c = \pm e_0$, and the norm is $0$ or $2\pi$ for $c = e_0$ and $\pi$ for $c = -e_0$, with the direction free. The identities between powers are the corresponding identities inside the commutative associative algebra $\mathbb{R}[\tilde o]$. The failure of the multiplicativity of the logarithm, $\log(\tilde o\tilde p)\neq\log \tilde o+\log \tilde p$ in general, is not a defect of the definition but the failure of $\exp(a)\exp(b) = \exp(a+b)$ for non-commuting arguments, treated in the next section; it already fails for the quaternions, and the octonionic case adds no new obstruction. Theorem. Let $a,b\in\mathbb{O}$. The subalgebra generated by $a$ and $b$ together with $e_0$ is associative, by Artin's theorem, and in it the following hold. Proof. (1) When $a$ and $b$ commute, the binomial theorem in the associative algebra generated by $a,b$ expands $(a+b)^n$ and the classical exponential proof applies term by term, with absolute convergence justifying the rearrangement; the algebra generated is associative by Artin's theorem, as in Octonion Algebra. (2) and (3) are the algebraic identities $[a,b]\neq0$ and $[a,b,c]\neq0$: the binomial expansion of $(a+b)^n$ acquires commutator corrections, and the product of the two exponential series acquires both commutator and associator corrections, as in the product rule of Octonion Analysis; explicit numerical examples are obtained by taking random non-commuting $a,b$ and truncating the series. The theorem states precisely how much of the elementary theory survives: all the functions of one variable behave as in the commutative associative case, because they live in the two-dimensional algebra $\mathbb{R}[\tilde o]$; and the exponential law, which is a statement about two variables, requires commutativity and uses associativity. This is the octonionic form of the classical situation for the quaternions, with the additional point that the subalgebra generated by any two octonions is associative but need not be commutative, so that failure of the exponential law in the octonionic case is caused by the same commutator obstruction as for the quaternions and by no new obstruction. Definition. A function $f:\mathbb{O}\to\mathbb{O}$ is axial if $f(\tilde o) = u(o_0,\lvert \tilde o\rvert) + v(o_0,\lvert \tilde o\rvert)\hat{\mathbf{o}}$ with $\hat{\mathbf{o}} = \mathbf{o}/\lvert\mathbf{o}\rvert$ and real-valued $u,v$; it is Fueter-regular if it is axial and monogenic with respect to $D$ away from the real axis, that is, if $Df = 0$. Theorem (Fueter–Sce). Let $F$ be holomorphic on a domain of the upper half-plane and let $\tilde F(\tilde o) = F(o_0 + i\lvert\mathbf{o}\rvert)$ be its axial extension to $\mathbb{O}$ in the sense that the imaginary part is read in the direction $\hat{\mathbf{o}}$. Then for the octonions, whose vector part has dimension $n = 7$, the function $$
f = \Delta_8^{(n-1)/2}\tilde F = \Delta_8^{3}\tilde F
$$ is axial and Fueter-regular away from the real axis; and conversely every Fueter-regular function on a domain invariant under the rotations of $\operatorname{Im}\mathbb{O}$ arises from a holomorphic function by this construction, up to the standard modifications of the boundary cases. In particular the fundamental solution $E$ is Fueter-regular, with axial data $u = 0$, $v = \lvert \tilde o\rvert^{-8}/\omega_7$, in agreement with the radial reduction of Octonion Analysis. Proof. The construction is the Fueter–Sce theorem for Clifford analysis with vector part of dimension $n$: a power $\Delta^{(n-1)/2}$ of the Laplacian applied to the axial extension of a holomorphic function produces a monogenic function, and the proof proceeds by expressing the operator $D$ in axial coordinates and showing that it lowers the monogenic degree, so that sufficiently many applications of $\Delta$ land in the kernel of $D$. The case $n = 7$ gives the exponent $3$. The details are in the sources cited; the assertion about $E$ is the radial reduction of Octonion Analysis. The construction is the source of the special functions of the octonion system: every Fueter-regular function is obtained from a holomorphic function of one complex variable, so the octonionic special functions with the correct symmetry form a class no larger than the classical ones, and the transcendence of the octonions affects the class only through the symmetry restriction. This is exactly the situation for the quaternions, where the analogue is the Fueter construction with the group $SO(3)$ acting on the imaginary part; the octonionic case replaces $SO(3)$ by $SO(7)$ acting on $\operatorname{Im}\mathbb{O}$, of which the automorphism group $G_2$ is a subgroup. Definition. For $\tilde o\in\mathbb{O}$ with $o_0 = \operatorname{Sc}\tilde o>0$, the gamma function is $$
\Gamma(\tilde o) = \int_0^{\infty}t^{\tilde o-e_0}e^{-t}\,dt ,
$$ the integrand being evaluated in the commutative associative algebra $\mathbb{R}[\tilde o]$, so that $t^{\tilde o-e_0} = \exp((\tilde o-e_0)\ln t)$ and $e^{-t}$ are real-analytic in $t$ with values in $\mathbb{R}[\tilde o]$; the beta function is $B(\tilde o,\tilde p) = \Gamma(\tilde o)\Gamma(\tilde p)/\Gamma(\tilde o+\tilde p)$ for $\tilde o,\tilde p$ with positive scalar parts. Theorem. The integral converges absolutely for $o_0>0$, defines a function of class $C^1$ in the eight coordinates, and satisfies the recurrence $\Gamma(\tilde o+e_0) = \tilde o\Gamma(\tilde o)$; for real $\tilde o = te_0$ it is the classical gamma function, and in general the value lies in the commutative associative algebra $\mathbb{R}[\tilde o]$ generated by the argument. The identities of the classical theory that involve two arguments hold when the two arguments commute, that is, when they lie in a common $\mathbb{R}[\tilde o]$, and fail in general, because they are obtained by manipulations of products that require associativity and commutativity. Proof. Convergence is the decay of $e^{-t}$ against the growth of $t^{o_0-1}$, componentwise in the algebra $\mathbb{R}[\tilde o]$, in which the integrand takes its values; the recurrence is the integration by parts inside $\mathbb{R}[\tilde o]$, which is legitimate there; the two-argument identities are the classical manipulations carried out in the subalgebra generated by the two arguments, which is associative by Artin's theorem but commutative only when the arguments commute, and those manipulations produce products that do not reassociate when the arguments do not commute. The same pattern gives the octonionic zeta function: for $\tilde o$ in $\mathbb{R}[\tilde o]$ with $o_0>1$ the series $\zeta(\tilde o) = \sum_{n\geq1}n^{-\tilde o}$ converges absolutely and defines a function of the single variable, and for two variables the Euler product and the functional equation require the commutativity of the products involved. The higher transcendental functions of the octonion system are therefore of two kinds, and the distinction is structural: functions of one octonion variable, which are as well behaved as in the complex case; and functions of several octonion variables, which inherit the commutator and associator corrections of the product and are correspondingly restricted. The special functions of the octonion system are of two kinds. The functions of a single variable — the exponential, the trigonometric and hyperbolic functions, the logarithm, the powers, the gamma function, the zeta function — are defined by one-variable power series or integrals whose terms lie in the commutative associative algebra $\mathbb{R}[\tilde o]$ generated by the argument, and they satisfy the classical identities unchanged: $\sin^2+\cos^2 = e_0$, $\cosh^2-\sinh^2 = e_0$, $\cos(2\tilde o) = \cos^2\tilde o-\sin^2\tilde o$, $\exp(\tilde o)\exp(-\tilde o) = e_0$, $\Gamma(\tilde o+e_0) = \tilde o\Gamma(\tilde o)$. For imaginary $\tilde o$ the exponential takes the values $\exp \tilde o = \cos\lvert \tilde o\rvert+\hat{\mathbf{o}}\sin\lvert \tilde o\rvert$ of modulus one, so that every direction of $\operatorname{Im}\mathbb{O}$ supplies an imaginary unit. The functions of two or more variables inherit the failure of associativity, and the boundary is exactly the product rule. The exponential law $\exp(a)\exp(b) = \exp(a+b)$ holds when $ab = ba$ and fails otherwise; the bracketings of $\exp a\exp b\exp c$ differ by associators and no bracketing is preferred; the multiplicativity of the logarithm and of the powers fails; and the two-argument identities of the gamma and zeta functions hold only for commuting arguments. The functions that survive with full strength are the axial ones and their subclass the Fueter-regular functions, which are obtained from holomorphic functions of one complex variable by the Fueter–Sce construction and are invariant under the rotations of $\operatorname{Im}\mathbb{O}$, with the group $G_2$ of Octonions and the Exceptional Lie Groups acting as a symmetry group of the whole theory.The Exponential and the Elementary Functions
The Logarithm and Powers
The Failure of Associativity in the Special Functions
The Exponential Law
Axial Functions and the Fueter Construction
The Gamma Function and the Higher Functions
Summary
Summary of Notation
Symbol
Meaning
$\tilde o = o_0+\mathbf{o}$, $\hat{\mathbf{o}} = \mathbf{o}/\lvert\mathbf{o}\rvert$
Octonion, its imaginary direction
$\exp$, $\sin$, $\cos$, $\sinh$, $\cosh$
One-variable series of the octonion variable
$\log \tilde o$, $\tilde o^\alpha$
Multivalued logarithm, principal branch, powers
$\mathbb{R}[\tilde o]$
Commutative associative subalgebra generated by $\tilde o$
$\Gamma(\tilde o) = \int_0^\infty t^{\tilde o-e_0}e^{-t}dt$
Octonionic gamma function on $o_0>0$
$B(\tilde o,\tilde p)$, $\zeta(\tilde o)$
Beta and zeta functions, single-variable cases
$u(o_0,\lvert \tilde o\rvert)+v(o_0,\lvert \tilde o\rvert)\hat{\mathbf{o}}$
Axial function
$D = \sum_ke_k\partial_k$, Fueter-regular
Cauchy–Riemann operator; axial monogenic functions
$\Delta_8$
Laplacian of $\mathbb{R}^8$, used in the Fueter–Sce construction
$[\tilde o,\tilde p]$, $[\tilde o,\tilde p,\tilde r]$
Commutator and associator
Further Reading