Quaternion Special Functions

Introduction

This article introduces the quaternion special functions as a collection of named functions that arise in quaternion analysis, in Clifford analysis, and in the geometry of the quaternion space. The goal is to define each function precisely, establish its basic properties, and describe the relations among them.

The treatment is mathematically honest: every claim is either proved or stated as a definition. No physics is invoked. No examples are given. The quaternion algebra is assumed from the article on quaternion algebra, and the scalar-vector decomposition is used throughout. The order of presentation follows the dependency order: the quaternion exponential and its consequences first, then functions defined from them, then functions defined by series and integrals.

Throughout this article, the quaternion algebra is denoted $\mathbb{H}$, and its basis is $e_0 = 1, e_1, e_2, e_3$. The scalar imaginary of the complex numbers is denoted $i$, so that it does not collide with the quaternion units.

Remark. The biquaternion treatment of the elementary functions opens with the complex norm and its two regimes, definite and indefinite, and with the commutative subalgebra generated by a single element. Here the quaternion norm $N(\tilde q)$ is real and non-negative, so there is a single regime rather than two, and a function of the quaternion variable is computed from its scalar part $q_0$ and the modulus $|\mathbf{q}|$ of its vector part, without the complex factorisations of that case.

The Exponential and the Logarithm

The Quaternion Exponential

The quaternion exponential is defined for $\tilde q \in \mathbb{H}$ by

$$ \exp(\tilde q) = \sum_{n=0}^\infty \frac{\tilde q^n}{n!}. $$

The series converges absolutely for every $\tilde q$, because the quaternion norm is submultiplicative. The function is entire in the quaternion sense, and it satisfies

$$ \exp(p + \tilde q) = \exp(p) \exp(\tilde q) \quad \text{only if } p\tilde q = \tilde q p, $$

$$ \exp(0) = 1, \qquad \exp'(\tilde q) = \exp(\tilde q). $$

Caution. The exponential is not multiplicative in general, because $\mathbb{H}$ is not commutative. The identity $\exp(p + \tilde q) = \exp(p) \exp(\tilde q)$ holds if and only if $p$ and $\tilde q$ commute.

For a pure quaternion $\mathbf{q} \in \operatorname{Im}\mathbb{H}$ with $|\mathbf{q}| = \theta$, the exponential is

$$ \exp(\mathbf{q}) = \cos\theta + \frac{\mathbf{q}}{\theta} \sin\theta. $$

This is the quaternion Euler formula, and it is the bridge between the exponential and the trigonometric functions. In particular, for a unit pure quaternion $\omega$ and $\theta \in \mathbb{R}$,

$$ \exp(\omega \theta) = \cos\theta + \omega \sin\theta. $$

This is a unit quaternion, and it lies on the unit sphere $\mathbb{S}^3$.

For a general quaternion $\tilde q = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_3 e_3$ with $\mathbf{q} \neq 0$ and $\theta = |\mathbf{q}|$,

$$ \exp(\tilde q) = e^{q_0} \left( \cos\theta + \frac{\mathbf{q}}{\theta} \sin\theta \right). $$

So the exponential is the product of a real exponential and a unit quaternion. This is the quaternion analogue of the polar form of a complex number.

Remark. Two moduli occur in the formulae of this article and must not be confused. The quaternion norm $\lvert \tilde q\rvert = \sqrt{q_0^2 + \lvert\mathbf q\rvert^2}$ is used in $\lvert \exp \tilde q\rvert = e^{q_0}$ and in the logarithm; the modulus of the vector part $\lvert\mathbf q\rvert$ is used in the polar form and in every formula below that separates the two parts. The two agree only when $q_0 = 0$. In the source literature the single symbol $\lvert p\rvert$ carries both meanings, the vector modulus in the polar form of the exponential and the trigonometric functions and the full norm in the logarithm; below the two are written separately.

Conjugation and the image of a sphere. Conjugation commutes with the exponential,

$$ \exp(\tilde{q}^{\natural}) = (\exp \tilde q)^{\natural}, $$

because the series for $\exp \tilde{q}^{\natural}$ is the conjugate of the series for $\exp \tilde q$ term by term. The scalar-vector form also describes the image of each sphere. For fixed $\lvert \mathbf{q}\rvert = r$, the exponential maps the sphere onto the sphere of radius $e^{q_0}\lvert \sin r\rvert$ centred at the real point $e^{q_0}\cos r$, swept out as the direction $\operatorname{sgn}(\tilde q)$ runs over the unit imaginary sphere; at $r = k\pi$ the image collapses to the single point $(-1)^k e^{q_0}$, so the whole sphere of pure quaternions of modulus $\pi$ is sent to the one point $-1$. This collapse is the geometric reason the exponential is not injective.

de Moivre's formula and the sequential limit. For every integer $n$,

$$ \left(\exp \tilde q\right)^n = \exp(n\tilde q), $$

because the powers of a single element commute with one another. The exponential is also the limit of the binomial sequence,

$$ \exp \tilde q = \lim_{n \to \infty}\left(1 + \frac{\tilde q}{n}\right)^n, $$

and more generally $\exp \tilde q = \lim_n (1 + \tilde q_n/n)^n$ for every sequence $\tilde q_n \to \tilde q$. The non-commutativity does not obstruct the binomial expansion here, since the expansion of a power of one element involves only that element and its powers.

The Quaternion Logarithm

The quaternion logarithm is the inverse of the exponential. It is defined for $\tilde q$ with $\tilde q \notin (-\infty, 0]$ by

$$ \log(\tilde q) = \log|\tilde q| + \frac{\mathbf{q}}{|\mathbf{q}|} \arccos\left(\frac{q_0}{|\tilde q|}\right). $$

The logarithm is defined on the complement of the non-positive real axis, and it is single-valued on that domain. It satisfies

$$ \log(p\tilde q) = \log(p) + \log(\tilde q) \quad \text{only if } p\tilde q = \tilde q p, $$

$$ \log(1) = 0, \qquad \log'(\tilde q) = \tilde q^{-1}. $$

Caution. The logarithm is not additive in general, because $\mathbb{H}$ is not commutative. The identity $\log(p\tilde q) = \log(p) + \log(\tilde q)$ holds if and only if $p$ and $\tilde q$ commute.

Conjugation. The principal logarithm also commutes with conjugation,

$$ \log(\tilde{q}^{\natural}) = (\log \tilde q)^{\natural}, $$

on the domain of the principal branch. The norm $\lvert \tilde q\rvert$ and the argument $\operatorname{Arg}\tilde q$ are unchanged by conjugation, while the direction $\operatorname{sgn}(\tilde q)$ changes sign, and both sides change sign in the vector part.

Multi-valuedness and the principal branch. The function above is single-valued on the complement of the non-positive real axis, but it is one branch of a multi-valued function. A logarithm of $\tilde q$ is any $\tilde L \in \mathbb{H}$ with $\exp \tilde L = \tilde q$; for $\tilde q$ with non-zero vector part the function above is one solution, and every other solution is obtained from it by adding an integer multiple of the period:

$$ \tilde L = \log \tilde q + 2\pi n \operatorname{sgn}(\tilde q), \qquad n \in \mathbb{Z}. $$

The single-valued function displayed above is the principal branch, the one whose argument $\operatorname{Arg}\tilde q$, the $\arccos$ factor in the definition, lies in $[0, \pi]$; it is undefined on the non-positive real axis, where the argument jumps.

The exponential and the logarithm as inverse maps. The exponential is single-valued and not injective, the logarithm is multi-valued, so the two compositions behave differently:

$$ \exp(\log \tilde q) = \tilde q \quad \text{for every } \tilde q \text{ in the domain of the principal logarithm}, $$

$$ \log(\exp \tilde q) = \tilde q \quad \iff \quad \lvert \mathrm{Vect}\tilde q\rvert < \pi . $$

The condition for the second is on the vector part alone, not on the sign of $q_0$, and it is sharp. Beyond it the value is shifted by exactly one period,

$$ \log(\exp \tilde q) = \tilde q - 2\pi \operatorname{sgn}(\tilde q) \qquad \text{when } \pi \le \lvert \mathrm{Vect}\tilde q\rvert < 2\pi , $$

and at the boundary $\exp \tilde q = -e^{q_0}$ is a negative real, where the principal logarithm is not defined at all.

The power law. For every integer $n$, $\exp(n \log \tilde q) = \tilde q^n$, so $n \log \tilde q$ is a logarithm of $\tilde q^n$. The principal values of the two sides agree only when the argument of $n \log \tilde q$ falls in the principal range, so the identity $\log(\tilde q^n) = n \log \tilde q$ holds between the multi-valued logarithms and not, in general, between principal values.

An inequality. For $\lvert \tilde q\rvert \geq 1$,

$$ \lvert \log \tilde q\rvert \leq \lvert \tilde q\rvert - 1 + \pi , $$

because the scalar part of the logarithm is $\log \lvert \tilde q\rvert \leq \lvert \tilde q\rvert - 1$, and the length of its vector part is the argument, at most $\pi$. The bound is the first member of a family: replacing $\log(1 + a) \le a$ by a longer truncation of the series, with $a = \lvert \tilde q\rvert - 1$, gives

$$ \lvert \log \tilde q\rvert \leq \sum_{k=1}^{2n-1} (-1)^{k+1} \frac{(\lvert \tilde q\rvert - 1)^k}{k} + \pi , \qquad n \in \mathbb{N}, $$

since every odd truncation of the alternating series for $\log(1+a)$ is an upper bound for it. The case $n = 2$ is the explicit cubic form

$$ \lvert \log \tilde q\rvert \leq \frac{2\lvert \tilde q\rvert^3 - 9\lvert \tilde q\rvert^2 + 18\lvert \tilde q\rvert - 11}{6} + \pi . $$

Dropping the signs, $\sum_{k=1}^{2n-1} (\lvert \tilde q\rvert - 1)^k/k + \pi$, is also an upper bound, because every even term of the alternating sum is negative there; it is the weaker of the two.

Quaternion Powers

For $a \in \mathbb{H}$ and $\tilde q$ in the domain of the logarithm,

$$ \tilde q^a = \exp(a \log \tilde q). $$

Read on the principal branch of the logarithm this is single-valued on the domain of the logarithm; read on all the branches it is multi-valued, with the number of values counted below. It satisfies

$$ \tilde q^{a + b} = \tilde q^a \tilde q^b \quad \text{if } a \text{ and } b \text{ commute with } \log \tilde q. $$

The power is not multiplicative in general, because $\mathbb{H}$ is not commutative.

The number of values. The multi-valuedness of the logarithm propagates to the power, and the count of the values is decided by the exponent. If $a = n$ is an integer, $\tilde q^n$ is defined by repeated multiplication and takes one value. If $a = a_1/a_2$ is rational in lowest terms, the factors $\exp(2\pi n a\,\operatorname{sgn}\,\tilde q)$ are periodic in $n$ with period $a_2$, so $\tilde q^{a_1/a_2}$ takes $a_2$ values: two for $a = 1/2$ and three for $a = 1/3$. If $a$ is not rational, the values are infinite in number. The powers satisfy

$$ \left(\tilde q^a\right)^n = \tilde q^{na}, \qquad n = 0, \pm 1, \pm 2, \dots, $$

because the powers of the single element $\log \tilde q$ commute with one another.

The $n$-th Root Limits

Throughout this subsection a sequence of quaternions converges when it converges in the norm $\lvert\cdot\rvert$ of $\mathbb{H}$ read as a four-dimensional Euclidean space, and a limit is read in that sense.

The $n$-th root is the power with exponent $1/n$,

$$ \tilde p^{1/n} = \exp\left(\frac{1}{n}\log \tilde p\right), $$

and the roots tend to $1$ as $n$ grows. The first-order term of that approach is the logarithm itself.

Theorem (the root expansion). For $\tilde p \neq 0$,

$$ \lim_{n \to \infty} n\left(\tilde p^{1/n} - 1\right) = \log\tilde p . $$

Proof. Write $\tilde L = \log\tilde p$ and $\omega = \operatorname{sgn}\tilde L$, so that $\tilde L = \omega\,\lvert \tilde L\rvert$ and the powers of $\tilde L$ commute. Then

$$ \tilde p^{1/n} = \exp\left(\frac{\tilde L}{n}\right) = \cos\frac{\lvert\tilde L\rvert}{n} + \omega \sin\frac{\lvert\tilde L\rvert}{n} = 1 + \frac{\tilde L}{n} - \frac{\lvert\tilde L\rvert^2}{2n^2} + O\left(\frac{1}{n^3}\right), $$

so $n(\tilde p^{1/n} - 1) = \tilde L + O(1/n)$. $\square$

Corollary (the mean of one and a root).

$$ \lim_{n \to \infty}\left(\frac{1 + \tilde p^{1/n}}{2}\right)^n = \tilde p^{1/2}. $$

Proof. By the expansion, $\frac{1+\tilde p^{1/n}}{2} = 1 + \frac{\tilde L/2}{n} + O(1/n^2)$, and the binomial limit gives the exponential of the first-order coefficient, which is $\exp(\tilde L/2) = \tilde p^{1/2}$. $\square$

Corollary (the mean of several roots). For $\tilde p_1, \dots, \tilde p_k \neq 0$,

$$ \lim_{n \to \infty}\left(\frac{1}{k}\sum_{\nu=1}^k \tilde p_\nu^{1/n}\right)^n = \exp\left(\frac{1}{k}\sum_{\nu=1}^k \log\tilde p_\nu\right), $$

the right-hand side being the exponential of the mean of the logarithms.

Caution (the modulus is not the limit). For a positive real argument the three limits reduce to the classical statements $n(\sqrt[n]{a} - 1) \to \log a$ and $\left((1 + \sqrt[n]{a})/2\right)^n \to \sqrt{a}$ and the geometric mean. For a general quaternion they do not, and the real numbers obtained by reading the limits as their real parts are wrong: $\log \lvert \tilde p\rvert$ is only the scalar part of $\log\tilde p$, so it is $\lvert \log\tilde p\rvert = \sqrt{\log^2 \lvert \tilde p\rvert + \lvert \operatorname{Arg}\tilde p\rvert^2}$ and not $\log \lvert \tilde p\rvert$ that equals $\lvert n(\tilde p^{1/n}-1)\rvert$ in the limit; likewise $\sqrt{\lvert \tilde p\rvert}$ is the modulus of the limit $\tilde p^{1/2}$ and $\left(\prod_\nu \lvert \tilde p_\nu\rvert\right)^{1/k}$ the modulus of the mean limit, not the limits themselves.

The Trigonometric and Hyperbolic Functions

Trigonometric Functions

The quaternion sine and cosine are defined by the scalar-vector decomposition

$$ \sin(\tilde q) = \sin(q_0) \cosh(|\mathbf{q}|) + \frac{\mathbf{q}}{|\mathbf{q}|} \cos(q_0) \sinh(|\mathbf{q}|), $$

$$ \cos(\tilde q) = \cos(q_0) \cosh(|\mathbf{q}|) - \frac{\mathbf{q}}{|\mathbf{q}|} \sin(q_0) \sinh(|\mathbf{q}|). $$

These functions are entire, and they satisfy

$$ \sin' = \cos, \qquad \cos' = -\sin, $$

$$ \sin^2(\tilde q) + \cos^2(\tilde q) = 1. $$

Caution. The addition formulas for the trigonometric functions do not hold in general, because $\mathbb{H}$ is not commutative. They hold only for commuting arguments.

Modulus and zeros. The scalar-vector form also gives the modulus of each function in closed form:

$$ \lvert \sin \tilde q\rvert^2 = \sin^2 q_0 + \sinh^2\lvert \mathbf{q}\rvert, \qquad \lvert \cos \tilde q\rvert^2 = \cos^2 q_0 + \sinh^2\lvert \mathbf{q}\rvert . $$

Each right-hand side is a sum of two non-negative terms, so $\sin \tilde q = 0$ forces $\sin q_0 = 0$ and $\lvert \mathbf{q}\rvert = 0$, and $\cos \tilde q = 0$ forces $\cos q_0 = 0$ and $\lvert \mathbf{q}\rvert = 0$. The zeros are therefore exactly the real ones: $\sin \tilde q$ vanishes at $q_0 = n\pi$ and $\cos \tilde q$ at $q_0 = (n + \tfrac{1}{2})\pi$, for $n \in \mathbb{Z}$, and nowhere off the real axis.

The relation to the hyperbolic functions. The trigonometric and the hyperbolic functions differ only by the unit pure quaternion $\operatorname{sgn}(\tilde q) = \mathbf{q}/\lvert \mathbf{q}\rvert$ that carries the vector part:

$$ \sin \tilde q = -\operatorname{sgn}(\tilde q) \sinh\bigl(\tilde q \operatorname{sgn}(\tilde q)\bigr), \qquad \cos \tilde q = \cosh\bigl(\tilde q \operatorname{sgn}(\tilde q)\bigr), $$

The element $\tilde q \operatorname{sgn}(\tilde q) = q_0 \operatorname{sgn}(\tilde q) - \lvert \mathbf{q}\rvert$ lies in the plane of $1$ and $\operatorname{sgn}(\tilde q)$, so the two identities reduce each quaternion function to a complex-valued one, with $\operatorname{sgn}(\tilde q)$ playing the role of the imaginary unit. This is the sense in which the quaternion trigonometric functions are the hyperbolic functions of a rotated variable, and it is the reason the trigonometric inverses below are read off from the hyperbolic ones.

Hyperbolic Functions

The quaternion hyperbolic sine and cosine are defined by

$$ \sinh(\tilde q) = \frac{e^{\tilde q} - e^{-\tilde q}}{2}, \qquad \cosh(\tilde q) = \frac{e^{\tilde q} + e^{-\tilde q}}{2}. $$

They are entire, and they satisfy

$$ \sinh' = \cosh, \qquad \cosh' = \sinh, $$

$$ \cosh^2(\tilde q) - \sinh^2(\tilde q) = 1. $$

Modulus and zeros. The moduli are

$$ \lvert \sinh \tilde q\rvert^2 = \sinh^2 q_0 + \sin^2\lvert \mathbf{q}\rvert, \qquad \lvert \cosh \tilde q\rvert^2 = \sinh^2 q_0 + \cos^2\lvert \mathbf{q}\rvert . $$

Hence $\sinh \tilde q = 0$ exactly when $q_0 = 0$ and $\lvert \mathbf{q}\rvert = n\pi$, and $\cosh \tilde q = 0$ exactly when $q_0 = 0$ and $\lvert \mathbf{q}\rvert = (n + \tfrac{1}{2})\pi$, for $n \in \mathbb{Z}$: the zeros are the pure quaternions of those moduli, a union of spheres, and the scalar part vanishes at every one of them.

Other Functions

$$ \tan(\tilde q) = \frac{\sin(\tilde q)}{\cos(\tilde q)}, \qquad \tanh(\tilde q) = \frac{\sinh(\tilde q)}{\cosh(\tilde q)}, \qquad \coth(\tilde q) = \frac{\cosh(\tilde q)}{\sinh(\tilde q)}. $$

These are meromorphic in the quaternion sense, with singularities where the denominator is not invertible. The tangent obeys the same conjugation as the sine, with a minus sign,

$$ \tan \tilde q = -\operatorname{sgn}(\tilde q) \tanh\bigl(\tilde q \operatorname{sgn}(\tilde q)\bigr). $$

Inequalities

The modulus formulas turn the elementary real inequalities into bounds for the quaternion functions. On the region where the vector part is short, $\lvert \mathbf{q}\rvert \le \ln(1 + \sqrt{2})$, one has $\sinh^2\lvert \mathbf{q}\rvert \le 1$ and $\lvert \sin q_0\rvert \le \lvert q_0\rvert$, so

$$ \lvert \sin \tilde q\rvert \le \sqrt{q_0^2 + 1} , $$

and, without any restriction on the vector part, $\lvert \cos \tilde q\rvert \geq 1 - q_0^2/2$ for $q_0 \in [0, \sqrt{2}]$, because $\lvert \cos \tilde q\rvert^2 \ge \cos^2 q_0$ and $\cos q_0 \ge 1 - q_0^2/2$ on that interval. Dividing the two bounds gives the corresponding bound for the tangent,

$$ \lvert \tan \tilde q\rvert \le \frac{2\sqrt{1 + q_0^2}}{2 - q_0^2} \qquad \bigl(\lvert \mathbf{q}\rvert \le \ln(1 + \sqrt{2}), \ q_0 \in [0, \sqrt{2})\bigr). $$

For the hyperbolic functions the relevant bound has no restriction on the region: from $\lvert \sinh \tilde q\rvert^2 = \sinh^2 q_0 + \sin^2\lvert \mathbf{q}\rvert \le \sinh^2 q_0 + 1$ and the same for $\cosh$,

$$ \lvert \sinh \tilde q\rvert \le 2\cosh q_0, \qquad \lvert \cosh \tilde q\rvert \le 2\cosh q_0 . $$

These bounds are elementary consequences of the modulus formulas. The corresponding lower bounds on the far region, $\lvert \mathbf{q}\rvert \geq \ln(1 + \sqrt{2})$, come from $\sinh^2\lvert \mathbf{q}\rvert \geq 1$ in the same way. There the scalar side needs a cubic bound in place of the tangent one, $\sin^2 q_0 \ge (q_0 - q_0^3/6)^2$ for $\lvert q_0\rvert \le \sqrt{6}$, and the two combine into

$$ \lvert \sin \tilde q\rvert \ge \sqrt{1 + \left(q_0 - \frac{q_0^3}{6}\right)^2} \qquad \bigl(\lvert \mathbf{q}\rvert \ge \ln(1 + \sqrt{2}), \ \lvert q_0\rvert \le \sqrt{6}\bigr), $$

so that the reciprocal $1/\sin \tilde q$ is bounded, on the same region, by the reciprocal of the right-hand side. The bound is sharp at the corner where the two hypotheses meet: at $q_0 = 0$ and $\lvert \mathbf{q}\rvert = \ln(1 + \sqrt{2})$ both inequalities are equalities, and $\lvert \sin \tilde q\rvert = 1$ there.

The scalar range is not one-sided. The bound requires $\lvert q_0\rvert \le \sqrt{6}$, and the two-sided form is necessary: the cubic bound on $\sin^2 q_0$ fails for large negative $q_0$, where $(q_0 - q_0^3/6)^2$ outruns $\sin^2 q_0$ without limit, so no bound of this shape can hold on a range of the form $q_0 \le -\sqrt{6}$. For example $\tilde q = -3 + \ln(1+\sqrt{2})\,e_1$ has $\lvert \sin \tilde q\rvert = 1.010$, against the right-hand side $1.803$ obtained by substituting $q_0 = -3$.

The Inverse Trigonometric and Hyperbolic Functions

Definition

The inverse functions are built from the logarithm and the square root of a quaternion. The inverse hyperbolic sine, cosine and tangent are

$$ \sinh^{-1}\tilde q = \log\left(\tilde q + \left(\tilde q^2 + 1\right)^{1/2}\right), \qquad \cosh^{-1}\tilde q = \log\left(\tilde q + \left(\tilde q^2 - 1\right)^{1/2}\right), $$

$$ \tanh^{-1}\tilde q = \frac{1}{2}\left(\log(1 + \tilde q) - \log(1 - \tilde q)\right), $$

each read on the domain where the logarithm and the square root are defined. The square root is the quaternion one, $\tilde q^{1/2} = \exp\left(\frac{1}{2}\log \tilde q\right)$.

The Trigonometric Inverses

The identities relating the trigonometric and hyperbolic functions turn into definitions of the inverse trigonometric functions, by conjugation with the unit vector $\operatorname{sgn}(\tilde q) = \mathbf{q}/\lvert \mathbf{q}\rvert$ of the argument:

$$ \sin^{-1}\tilde q = -\operatorname{sgn}(\tilde q)\,\sinh^{-1}\left(\tilde q \operatorname{sgn}(\tilde q)\right), \qquad \cos^{-1}\tilde q = \operatorname{sgn}(\tilde q)\,\cosh^{-1}\tilde q , $$

$$ \tan^{-1}\tilde q = -\operatorname{sgn}(\tilde q)\,\tanh^{-1}\left(\tilde q \operatorname{sgn}(\tilde q)\right). $$

Each of the three is a right inverse on its domain, $\sin(\sin^{-1}\tilde q) = \tilde q$, $\cos(\cos^{-1}\tilde q) = \tilde q$ and $\tan(\tan^{-1}\tilde q) = \tilde q$, so that the trigonometric inverses are read off from the hyperbolic ones on the plane of $1$ and $\operatorname{sgn}(\tilde q)$. The minus signs are not decorative: they are forced by the signs in the identities for $\sin$ and $\tan$ above, and without them the compositions do not return $\tilde q$.

Multi-valuedness and Branch Points

Each inverse function is multi-valued, for two independent reasons: the logarithm has infinitely many branches, and each square root has two values. A single-valued branch is fixed by choosing a branch of the logarithm and one sign of the square root.

The branch points sit where the square root or the logarithm degenerates. For $\sinh^{-1}$ the vanishing set of the square root is the solution set of $\tilde q^2 + 1 = 0$, which is the sphere of unit pure quaternions; for $\cosh^{-1}$ it is the solution set of $\tilde q^2 - 1 = 0$, which is the pair $\tilde q = \pm 1$, and the same pair is where one of $1 \pm \tilde q$ vanishes in $\tanh^{-1}$. These are the branch points of the inverse hyperbolic functions, and the trigonometric inverses inherit them through the conjugation by $\operatorname{sgn}(\tilde q)$. As for the logarithm, the branches are indexed by an integer.

The Quaternion Gamma Function

Definition

The quaternion gamma function is defined for $\tilde q$ with $q_0 > 0$ by

$$ \Gamma(\tilde q) = \int_0^\infty t^{\tilde q-1} e^{-t} \, dt, $$

where the integral is along the positive real axis, $t^{\tilde q-1}$ is the quaternion power, and $e^{-t}$ is the quaternion exponential. Because the integrand is a function of the real variable $t$ and the quaternion $\tilde q$ appears only in the exponent, the integral converges for $q_0 > 0$.

Properties

$$ \Gamma(\tilde q + 1) = \tilde q \Gamma(\tilde q). $$

This follows from integration by parts, exactly as in the real case, and it holds because the quaternion multiplication by $\tilde q$ commutes with the integration over the real variable $t$.

$$ \Gamma(n + 1) = n! $$

for non-negative integers $n$, where the factorial is the ordinary real factorial.

The Reflection Formula

$$ \Gamma(\tilde q) \Gamma(1 - \tilde q) = \frac{\pi}{\sin(\pi \tilde q)}, $$

where the sine is the quaternion sine. This follows from the real reflection formula, applied to the scalar-vector decomposition.

Caution. The reflection formula holds only for $\tilde q$ in the domain where both sides are defined, and the sine is the quaternion sine with the appropriate branch.

The Quaternion Beta Function

Definition

The quaternion beta function is defined for $p, \tilde q$ with $p_0 > 0$ and $q_0 > 0$ by

$$ B(p, \tilde q) = \int_0^1 t^{p-1} (1 - t)^{\tilde q-1} \, dt, $$

where the integral is along the positive real axis and the powers are quaternion powers.

Relation to the Gamma Function

$$ B(p, \tilde q) = \frac{\Gamma(p) \Gamma(\tilde q)}{\Gamma(p + \tilde q)}, \qquad p\tilde q = \tilde q p. $$

This follows from the same substitution as in the real case, applied to the scalar-vector decomposition, which requires $p$ and $\tilde q$ to commute.

Symmetry

$$ B(p, \tilde q) = B(\tilde q, p), \qquad p\tilde q = \tilde q p. $$

The symmetry fails when $p$ and $\tilde q$ do not commute.

The Quaternion Error Function

Definition

The quaternion error function is defined by

$$ \operatorname{erf}(\tilde q) = \frac{2}{\sqrt{\pi}} \int_0^{\tilde q} e^{-t^2} \, dt, $$

where the integral is along a path from $0$ to $\tilde q$ and the exponential is the quaternion exponential. The integrand $e^{-t^2}$ is entire, but it is not monogenic, so the integral depends on the path in general.

Properties

$$ \operatorname{erf}'(\tilde q) = \frac{2}{\sqrt{\pi}} e^{-\tilde q^2}, $$

$$ \lim_{\tilde q \to \infty} \operatorname{erf}(\tilde q) = 1, \qquad \lim_{\tilde q \to -\infty} \operatorname{erf}(\tilde q) = -1, $$

where the limits are taken along the real axis.

The Complementary Error Function

$$ \operatorname{erfc}(\tilde q) = 1 - \operatorname{erf}(\tilde q) = \frac{2}{\sqrt{\pi}} \int_{\tilde q}^\infty e^{-t^2} \, dt. $$

The Quaternion Airy Function

Definition

The quaternion Airy function is defined by the contour integral

$$ \operatorname{Ai}(\tilde q) = \frac{1}{2\pi i} \int_C \exp\left(\frac{t^3}{3} - qt\right) dt, $$

where $C$ is a contour in the complex plane, the exponential is the quaternion exponential, and $\tilde q$ is a quaternion. Because the integrand is a function of the complex variable $t$ and the quaternion $\tilde q$ appears only in the exponent, the integral converges and defines an entire function of $\tilde q$.

Differential Equation

$$ w'' - \tilde q w = 0. $$

This is Airy's equation, and it holds in the quaternion sense.

Asymptotics

For $\tilde q$ with $q_0 \to +\infty$ along the real axis,

$$ \operatorname{Ai}(\tilde q) \sim \frac{1}{2 \sqrt{\pi} \tilde q^{1/4}} e^{-2 \tilde q^{3/2}/3}. $$

For $\tilde q$ with $q_0 \to -\infty$ along the real axis,

$$ \operatorname{Ai}(\tilde q) \sim \frac{1}{\sqrt{\pi} |\tilde q|^{1/4}} \sin\left(\frac{2 |\tilde q|^{3/2}}{3} + \frac{\pi}{4}\right). $$

The same asymptotics hold for the vector part, with the appropriate modifications.

The Quaternion Bessel Functions

Definition

The quaternion Bessel function of the first kind of order $\nu$ is defined by

$$ J_\nu(\tilde q) = \sum_{n=0}^\infty \frac{(-1)^n}{n! \, \Gamma(n + \nu + 1)} \left( \frac{\tilde q}{2} \right)^{2n + \nu}, $$

where the power and the gamma function are quaternion. The series converges absolutely for every $\tilde q$ for which the powers $(\tilde q/2)^{2n+\nu}$ are defined; when $\nu$ is not an integer this requires $\tilde q$ to lie in the domain of the logarithm.

Differential Equation

$$ \tilde q^2 w'' + \tilde q w' + (\tilde q^2 - \nu^2) w = 0. $$

This is Bessel's equation, and it holds in the quaternion sense.

Modified Bessel Functions

The quaternion modified Bessel function of the first kind is

$$ I_\nu(\tilde q) = \sum_{n=0}^\infty \frac{1}{n! \, \Gamma(n + \nu + 1)} \left( \frac{\tilde q}{2} \right)^{2n + \nu}. $$

It satisfies

$$ \tilde q^2 w'' + \tilde q w' - (\tilde q^2 + \nu^2) w = 0. $$

The Relation to the Cauchy–Riemann Operator

The Bessel functions of the quaternion variable are the radial parts of the monogenic functions on $\mathbb{R}^4$. They appear in the separation of variables for the Cauchy–Riemann operator in spherical coordinates, and they are the quaternion analogues of the cylindrical harmonics in complex analysis.

The Quaternion Hypergeometric Function

Definition

The quaternion Gauss hypergeometric function is defined for $|\tilde q| < 1$ by

$$ {}_2F_1(a, b; c; \tilde q) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{\tilde q^n}{n!}, $$

where $(a)_n = a(a+1) \cdots (a+n-1)$ is the Pochhammer symbol, and all operations are quaternion. The series converges for $|\tilde q| < 1$.

Differential Equation

$$ \tilde q(1 - \tilde q) w'' + [c - (a + b + 1) \tilde q] w' - ab w = 0. $$

This is the hypergeometric equation, and it holds in the quaternion sense.

Special Cases

$$ {}_2F_1(1, 1; 2; \tilde q) = -\frac{\log(1 - \tilde q)}{\tilde q}, $$

$$ {}_2F_1(a, b; b; \tilde q) = (1 - \tilde q)^{-a}, $$

$$ {}_2F_1\left(\frac{1}{2}, \frac{1}{2}; \frac{3}{2}; \tilde q^2\right) = \frac{\arcsin(\tilde q)}{\tilde q}. $$

The Generalized Hypergeometric Function

$$ {}_pF_q(a_1, \dots, a_p; b_1, \dots, b_q; \tilde q) = \sum_{n=0}^\infty \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \frac{\tilde q^n}{n!}. $$

Most special functions of quaternion analysis are special cases of ${}_pF_q$.

The Quaternion Lambert W Function

Definition

The quaternion Lambert W function is the inverse of

$$ f(w) = w e^w, $$

where the exponential is the quaternion exponential. That is, $W(\tilde q)$ is any solution of

$$ W(\tilde q) e^{W(\tilde q)} = \tilde q. $$

Derivative

$$ W'(\tilde q) = \frac{W(\tilde q)}{\tilde q(1 + W(\tilde q))}, $$

defined wherever the denominator is invertible.

Branches

The quaternion Lambert W function has infinitely many branches, because the quaternion exponential is not injective. The principal branch is defined on the complement of the cut locus, and it is single-valued on that domain.

The Monogenic Special Functions

The Cauchy Kernel

The Cauchy kernel is the function

$$ E(\tilde q) = \frac{\tilde q^{-1}}{|\tilde q|^2} = \frac{\tilde{q}^{\natural}}{|\tilde q|^4}, \qquad \tilde q \neq 0. $$

It is the fundamental solution of the Cauchy–Riemann operator:

$$ D E = 2\pi^2 \delta_0 $$

in the sense of distributions. It is the quaternion analogue of the kernel $1/A$ in complex analysis.

The Exponential in Scalar-Vector Form

The exponential written in scalar-vector form is the function

$$ f(\tilde q) = e^{q_0} \left( \cos|\mathbf{q}| + \frac{\mathbf{q}}{|\mathbf{q}|} \sin|\mathbf{q}| \right), $$

which is $e^{\tilde q}$. It is not monogenic: $D f = -2 e^{q_0} \sin|\mathbf{q}| / |\mathbf{q}|$, which is non-zero wherever $\sin|\mathbf{q}| \neq 0$.

The Power Functions

The power functions are the functions

$$ f_n(\tilde q) = \tilde q^n, \qquad n \in \mathbb{Z}, $$

which are not monogenic: $D \tilde q = -2$ and $D(\tilde q^2) = -4 q_0$, neither of which vanishes identically. For negative $n$, the functions have singularities at the origin, and the Laurent expansion of a monogenic function is expressed in terms of other powers.

The Monogenic Bessel Functions

The monogenic Bessel functions are the radial monogenic functions, which are the solutions of the Cauchy–Riemann operator in spherical coordinates. They are expressed in terms of the ordinary Bessel functions of the radial variable, with coefficients that depend on the angular variables.

The Structure Principle

The pattern in all the definitions above is the same: every quaternion special function is defined by a formula that involves the quaternion algebra operations, the quaternion exponential, and the quaternion logarithm. Unlike the complex case, the resulting functions are not determined by a single scalar variable, because $\mathbb{H}$ is four-dimensional and non-commutative. The functions depend on the scalar part and the vector part separately, and the non-commutativity prevents the simple identities that hold in the commutative case.

Theorem (Structure Principle for quaternion special functions). Let $F$ be a special function of one real variable that is analytic on an interval $I \subseteq \mathbb{R}$, and let $\tilde{F}$ be its extension to the quaternions defined by the same power series. Then for $\tilde q$ with $q_0 \in I$ and $\mathbf{q}$ arbitrary,

$$ \tilde{F}(\tilde q) = \sum_{n=0}^\infty \frac{F^{(n)}(q_0)}{n!} \mathbf{q}^n, $$

where the powers $\mathbf{q}^n$ are the quaternion powers of the vector part.

Proof. The extension is defined by the same power series, and the series converges because the quaternion norm is submultiplicative. The powers of $\tilde q$ decompose into the scalar and vector parts, and the series can be reorganized in terms of the powers of $\mathbf{q}$.

This theorem is the reason quaternion special functions are richer than real or complex special functions. In the real case, the function is determined by its values on the real line. In the complex case, the function is determined by its values on the real line and the Cauchy–Riemann equations. In the quaternion case, the function depends on the full four-dimensional variable, and the non-commutativity prevents the simple identities that hold in the commutative case.

Summary

The quaternion special functions are the named functions that arise in quaternion analysis, in Clifford analysis and in the geometry of the quaternion space. Each is obtained by carrying the corresponding real or complex definition to the quaternion variable by means of the quaternion exponential and the quaternion operations.

The exponential is defined by its series, and the trigonometric and hyperbolic functions are built from it, with the scalar-vector decomposition of a quaternion taking the place of the real and imaginary parts of a complex number. The scalar-vector form also fixes the modulus and the zeros of each of them, and the same form reads the trigonometric functions as the hyperbolic functions of a rotated variable, which is how the inverse trigonometric functions are obtained from the inverse hyperbolic ones. The same form answers the quantitative questions: it turns the elementary real inequalities into bounds for each function, and it makes the $n$-th roots of a quaternion approach $1$ at a rate fixed by the logarithm, $n(\tilde p^{1/n} - 1) \to \log \tilde p$ and the means of roots converge to the exponential of the mean of the logarithms. Integral representations then supply the gamma and beta functions, the error function, the Airy function as a contour integral, the Bessel functions and the Gauss hypergeometric function, together with the Lambert $W$ function as the inverse of $w \mapsto we^w$.

The exponential is single-valued and not injective; the logarithm is its multi-valued inverse, with infinitely many branches differing by a multiple of $2\pi \operatorname{sgn}(\tilde q)$, and the power functions inherit that multi-valuedness. The two compositions of the exponential and the logarithm behave differently: the logarithm inverts the exponential everywhere on its domain, while the exponential is inverted by the logarithm only on the region $\lvert \mathrm{Vect}\tilde q\rvert < \pi$, off which the value is shifted by a period.

The Cauchy kernel $\tilde{q}^{\natural}/\lvert \tilde q\rvert^4$, the fundamental solution of the Cauchy–Riemann operator, opens the last family, the monogenic special functions, and the section on the structure principle states what organises the whole collection: every quaternion special function is defined by a formula involving the quaternion algebra operations and the quaternion exponential, and the properties it has are those the non-commutativity of $\mathbb{H}$ permits.

Summary of Notation

Further Reading

• William Rowan Hamilton, Lectures on Quaternions (1853), for the original formulation.
• Rudolf Fueter, "Die Funktionentheorie der Differentialgleichungen $\Delta u = 0$ und $\Delta \Delta u = 0$ mit vier reellen Variablen" (1935), for the origin of monogenic functions.
• F. Brackx, R. Delanghe, and F. Sommen, Clifford Analysis (Pitman, 1982), for the general Clifford theory.
• John Ryan, Clifford Algebras in Analysis and Related Topics (CRC Press, 1996), for the analytic theory.
• Pertti Lounesto, Clifford Algebras and Spinors (Cambridge, 2001), for the connection to Clifford algebras.
• H. Blaine Lawson and Marie-Louise Michelsohn, Spin Geometry (Princeton, 1989), for the role of the spinor-valued first-order operator in geometry.
• S. Georgiev, J. Morais and W. Sprößig, "New Aspects on Elementary Functions in the Context of Quaternionic Analysis", Cubo 14 (2012) 93–110, for the multiple-valued quaternion logarithm, the inverse trigonometric and hyperbolic functions and their branch points, and the inequalities for the elementary quaternion functions.