Biquaternion Elementary Functions

Introduction

This article introduces the elementary functions of a biquaternion variable. It follows the basic algebra article, which defined the biquaternion algebra $\mathbb{B}$, its conjugations, and its six distinguished subspaces. The goal is to define the exponential, the trigonometric and hyperbolic functions, the logarithm, and the power functions, and to compute them in closed form. The exponential is treated in full: its closed form, its exponential coordinates at the identity, its group law, its kernel, and the exponentials in the two real directions of the trace-free part. Its Lie-group consequences — the group of units and the norm-one group — are in Biquaternion Lie Group and Exponential Structure.

The treatment is purely mathematical. No physics is invoked. No examples are given, except for the explicit counterexamples that establish the failure of the addition formulas in the non-commutative setting. The biquaternion algebra $\mathbb{B}$ is assumed from the basic algebra article, together with its scalar-vector decomposition, its biquaternion norm, and its four conjugations.

The key structural fact is that the elementary functions of a biquaternion are determined by the powers of the biquaternion, and the powers simplify dramatically in two cases: when the vector part has a nonzero complex norm, and when the vector part is nilpotent. These two cases cover all possibilities, and they lead to two regimes: the oscillatory regime and the nilpotent regime. The purpose of this article is to develop the elementary functions in both regimes.

The higher special functions — Bessel, hypergeometric, gamma, zeta, and the functions arising from the analysis — are treated in the companion article on biquaternion higher special functions. Throughout, a biquaternion is written

$$ \tilde{Q} = Q_0 e_0 + Q_1 e_1 + Q_2 e_2 + Q_3 e_3, \qquad Q_\mu \in \mathbb{C}, $$

or, more compactly, as

$$ \tilde{Q} = Q_0 e_0 + \mathbf{Q}, \qquad \mathbf{Q} = Q_1 e_1 + Q_2 e_2 + Q_3 e_3. $$

The scalar part is $Q_0 \in \mathbb{C}$; the vector part is $\mathbf{Q}$. The quaternion conjugate is $\tilde{Q}^{\natural} = Q_0 e_0 - \mathbf{Q}$, and the biquaternion norm is

$$ N(\tilde{Q}) = \tilde{Q} \tilde{Q}^{\natural} = Q_0^2 + Q_1^2 + Q_2^2 + Q_3^2. $$

The scalar imaginary is $i$, which commutes with the quaternion units.

The Complex Norm and the Two Regimes

The complex norm of the vector part $\mathbf{Q}$ is a fixed square root $$ B = \sqrt{Q_1^2 + Q_2^2 + Q_3^2} $$ of the value $(\mathbf{Q}, \mathbf{Q}) = Q_1^2 + Q_2^2 + Q_3^2$ of the complex quadratic form on the vector part. The form and its polarisation are owned by Biquaternion Norm and Invertibility, §The Polarisation and the Complex Quadratic Space; the square root $B$ and the subalgebra it generates are in Biquaternion Spectral Theory, §The Biquaternion Spectrum and §The Commutative Subalgebra Generated by a Single Biquaternion.

It is a complex number in general and plays the role of the "magnitude" of $\mathbf{Q}$. When $B \neq 0$ the vector part can be written $$ \mathbf{Q} = B \hat{n}, \qquad \hat{n} = \frac{\mathbf{Q}}{B}, $$ and $\hat{n}$ is a root of $-1$: since a pure biquaternion squares to $-\left(\mathbf{Q},\mathbf{Q}\right)e_0$, we have $\hat{n}^2 = \mathbf{Q}^2/B^2 = -e_0$. The choice of square root is a convention: replacing $B$ by $-B$ replaces the axis $\hat{n}$ by $-\hat{n}$, leaving both $\hat{n}^2$ and $\tilde{Q} = Q_0 e_0 + B\hat{n}$ unchanged.

The biquaternion norm is $$ N(\tilde{Q}) = \tilde{Q} \tilde{Q}^{\natural} = Q_0^2 + Q_1^2 + Q_2^2 + Q_3^2 = Q_0^2 + B^2, $$ the sum of the square of the scalar part and the square of the complex norm; this is the biquaternion analogue of $|A|^2 = a^2 + b^2$ for $A = a + bi$, with $a$ replaced by $Q_0$ and $b$ by $B$.

The Two Regimes

The vector part $\mathbf{Q}$ is a zero divisor exactly when $(\mathbf{Q}, \mathbf{Q}) = 0$, i.e. when $B = 0$; the criterion is in Biquaternion Zero Divisors, §Pure Zero Divisors. So there are two cases:

• Oscillatory regime: $B \neq 0$. The vector part is not a zero divisor, and it can be normalized to a root of $-1$. The powers of $\mathbf{Q}$ alternate, as in the quaternion case.
• Nilpotent regime: $B = 0$. The vector part is nilpotent, and $\mathbf{Q}^2 = 0$. The powers of $\mathbf{Q}$ truncate after the first.

These two cases cover all possibilities for the vector part. The elementary functions are computed separately in each case, and the formulas are different in the two regimes.

The Case $\mathbf{Q} = 0$

If $\mathbf{Q} = 0$, the biquaternion is a complex scalar $\tilde{Q} = Q_0 e_0$, and the elementary functions reduce to the ordinary complex elementary functions. This is a degenerate subcase of both regimes, since $B = 0$ and $\mathbf{Q} = 0$. It is treated separately in each section.

The Case $B = 0$, $\mathbf{Q} \neq 0$

In the nilpotent regime with a nonzero vector part, the element $\mathbf{Q}$ is a pure zero divisor, and its square vanishes. The elementary functions truncate, and the formulas are linear in $\mathbf{Q}$.

The Subalgebra Generated by a Single Biquaternion

The elementary functions of a single biquaternion are tractable because the powers of one element span a commutative subalgebra. The $\mathbb{C}$-linear span of $\{e_0, \tilde{Q}, \tilde{Q}^2, \dots\}$ is the commutative subalgebra generated by $\tilde{Q}$, written $\mathbb{C}[\tilde{Q}]$; it is the smallest subalgebra of $\mathbb{B}$ containing $e_0$ and $\tilde{Q}$.

Cayley–Hamilton. Direct computation from the product formula gives $\tilde{Q}^2 = Q_0^2 e_0 + 2Q_0\mathbf{Q} - (\mathbf{Q},\mathbf{Q})e_0$, that is $$ \tilde{Q}^2 - 2 Q_0 \tilde{Q} + N(\tilde{Q}) e_0 = 0. $$ Hence every power $\tilde{Q}^n$ with $n \geq 2$ is a $\mathbb{C}$-linear combination of $e_0$ and $\tilde{Q}$, and $\mathbb{C}[\tilde{Q}] = \operatorname{span}_{\mathbb{C}}\{e_0, \tilde{Q}\}$, of dimension two when $B \neq 0$ or $\mathbf{Q} \neq 0$, and one (equal to $\mathbb{C}e_0$) when $\mathbf{Q} = 0$.

Because $\mathbb{C}[\tilde{Q}]$ is commutative, the standard identities of complex analysis — the Pythagorean identity, the hyperbolic identity, the double-angle formulas, the relation between the exponential and the trigonometric functions — hold within it, and every elementary function $F(\tilde{Q})$ defined by a power series with coefficients in $\mathbb{C}$ lies in $\mathbb{C}[\tilde{Q}]$. This is the precise sense in which the non-commutativity of $\mathbb{B}$ does not obstruct the standard identities for a single biquaternion variable. The structure of $\mathbb{C}[\tilde{Q}]$ — its minimal polynomial, its two isomorphism types and the convergence of power series in it — is developed in Biquaternion Spectral Theory, §The Commutative Subalgebra Generated by a Single Biquaternion.

The Exponential

Definition

The exponential of a biquaternion $\tilde{Q}$ is defined by the power series

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

The series converges for every $\tilde{Q} \in \mathbb{B}$, because $\mathbb{B}$ is finite-dimensional and the Euclidean norm grows at most exponentially with $n$. So the exponential is an entire function on $\mathbb{B}$.

Reduction to the Vector Part

Write $\tilde{Q} = Q_0 e_0 + \mathbf{Q}$. The scalar part $Q_0 e_0$ commutes with everything, so by the identity $\exp(A + B) = \exp(A)\exp(B)$ for commuting $A$ and $B$,

$$ \exp(\tilde{Q}) = \exp(Q_0 e_0) \exp(\mathbf{Q}) = e^{Q_0} \exp(\mathbf{Q}), $$

where $e^{Q_0}$ is the ordinary complex exponential. So it suffices to compute the exponential of the vector part $\mathbf{Q}$.

The Case $B \neq 0$

If $B \neq 0$, write $\mathbf{Q} = B \hat{n}$ with $\hat{n}^2 = -e_0$. The powers of $\mathbf{Q}$ satisfy

$$ \mathbf{Q}^{2m} = (-1)^m B^{2m} e_0, \qquad \mathbf{Q}^{2m+1} = (-1)^m B^{2m+1} \hat{n}. $$

Separating even and odd powers in the exponential,

$$ \exp(\mathbf{Q}) = \left(\sum_{m=0}^{\infty} \frac{(-1)^m B^{2m}}{(2m)!}\right) e_0 + \left(\sum_{m=0}^{\infty} \frac{(-1)^m B^{2m+1}}{(2m+1)!}\right) \hat{n}. $$

These are the Taylor series of the complex cosine and sine:

$$ \exp(\mathbf{Q}) = \cos B \, e_0 + \sin B \, \hat{n}. $$

So

$$ \exp(\tilde{Q}) = e^{Q_0} \bigl(\cos B \, e_0 + \sin B \, \hat{n}\bigr), \qquad B \neq 0. $$

The Case $B = 0$

If $B = 0$, then $\mathbf{Q}^2 = 0$, so

$$ \mathbf{Q}^n = 0 \quad \text{for all } n \geq 2. $$

The power series of the exponential truncates to

$$ \exp(\mathbf{Q}) = e_0 + \mathbf{Q}. $$

So

$$ \exp(\tilde{Q}) = e^{Q_0} (e_0 + \mathbf{Q}), \qquad B = 0. $$

This is the analogue of the exponential of a nilpotent matrix: $\exp(N) = I + N$.

The Case $\mathbf{Q} = 0$

If $\mathbf{Q} = 0$, the biquaternion is a complex scalar, and both formulas reduce to

$$ \exp(\tilde{Q}) = e^{Q_0} e_0, $$

which is the ordinary complex exponential.

Summary of the Exponential

$$ \exp(\tilde{Q}) = \begin{cases} e^{Q_0} \bigl(\cos B \, e_0 + \sin B \, \hat{n}\bigr) & \text{if } B \neq 0, \\[2mm] e^{Q_0} (e_0 + \mathbf{Q}) & \text{if } B = 0, \end{cases} $$

where $B = \sqrt{Q_1^2 + Q_2^2 + Q_3^2}$ and $\hat{n} = \mathbf{Q}/B$.

Properties

Multiplicativity. The exponential satisfies $\exp(\tilde{P} + \tilde{Q}) = \exp(\tilde{P}) \exp(\tilde{Q})$ if $\tilde{P}$ and $\tilde{Q}$ commute. In general, the exponential is not multiplicative.

Non-vanishing. The exponential is never zero. In the oscillatory case, $e^{Q_0} \neq 0$ and the factor $\cos B \, e_0 + \sin B \, \hat{n}$ is invertible: its inverse is $\cos B \, e_0 - \sin B \, \hat{n}$, as follows from the identity

$$ (\cos B \, e_0 + \sin B \, \hat{n})(\cos B \, e_0 - \sin B \, \hat{n}) = (\cos^2 B + \sin^2 B) e_0 = e_0. $$

In the nilpotent case, $e^{Q_0} \neq 0$ and $(e_0 + \mathbf{Q})(e_0 - \mathbf{Q}) = e_0 - \mathbf{Q}^2 = e_0$.

Derivative in the scalar direction. The exponential satisfies $\partial_0 \exp(\tilde{Q}) = \exp(\tilde{Q})$, where $\partial_0 = \partial/\partial Q_0$ is the derivative in the scalar direction. This is the reason the exponential is the fundamental solution of the scalar differential operator $\partial_0$.

Derivative in the vector directions. The derivatives $\partial_k \exp(\tilde{Q})$ for $k = 1, 2, 3$ are not equal to $\exp(\tilde{Q}) e_k$ or $e_k \exp(\tilde{Q})$ in general. The reason is that $\partial_k \tilde{Q} = e_k$ does not commute with $\tilde{Q}$ unless $\tilde{Q}$ lies in the subalgebra generated by $e_0$ and $e_k$. The correct expression is the biquaternion analogue of the matrix identity $\partial e^M = \int_0^1 e^{sM}(\partial M)e^{(1-s)M}\,ds$, namely

$$ \partial_k \exp(\tilde{Q}) = \int_0^1 e^{s \tilde{Q}} e_k e^{(1-s)\tilde{Q}} \, ds. $$

This reduces to $\exp(\tilde{Q}) e_k$ only when $e_k$ commutes with $\tilde{Q}$.

Biquaternion norm. The biquaternion norm of the exponential is

$$ N(\exp(\tilde{Q})) = \exp(2 Q_0) = \exp(\mathrm{Tr}(\tilde{Q})), $$

where $\mathrm{Tr}(\tilde{Q}) = 2\,\mathrm{Sc}\,\tilde{Q} = 2Q_0$ is the trace of the element. The verification is direct: in the oscillatory regime, $N(\exp(\tilde{Q})) = e^{2 Q_0}(\cos^2 B + \sin^2 B) = e^{2 Q_0}$; in the nilpotent regime, $N(\exp(\tilde{Q})) = e^{2 Q_0}(1 + 0) = e^{2 Q_0}$.

Derivative at the Origin

The differential of the exponential at the origin is the identity, $d\exp_0 = \mathrm{id}$, so $\exp$ is a local diffeomorphism near $0$ and provides exponential coordinates near the identity of the group of units $\mathbb{B}^\times$ (the Lie-group side of this is in Biquaternion Lie Group and Exponential Structure).

The Group Law: $\exp(a)\exp(b)$ versus $\exp(a+b)$

The exponential is not a homomorphism from the additive group of the algebra to the multiplicative group of the units. Its failure to be one is measured by the bracket.

Commuting case. If $[a,b] = 0$, then

$$ \exp(a)\exp(b) = \exp(a+b) = \exp(b)\exp(a). $$

In particular, this holds when $a$ and $b$ lie in a common commutative subalgebra. For a single biquaternion $\tilde{Q}$, all elementary functions lie in the commutative subalgebra $\mathbb{C}[\tilde{Q}]$ generated by $e_0$ and $\tilde{Q}$, so the usual addition formula holds there.

General case. The Baker–Campbell–Hausdorff theorem gives, for sufficiently small $a, b$,

$$ \exp(a)\exp(b) = \exp\!\left(a + b + \frac{1}{2}[a,b] + \frac{1}{12}\bigl[a,[a,b]\bigr] - \frac{1}{12}\bigl[b,[a,b]\bigr] + \cdots\right), $$

a convergent series of iterated brackets. Thus $\exp(a)\exp(b)$ differs from $\exp(a+b)$ by the term $\tfrac{1}{2}[a,b]$ and higher brackets; the difference vanishes whenever $[a,b] = 0$, though the converse can fail, since a non-commuting pair may still satisfy $\exp(a)\exp(b) = \exp(a+b)$ when the BCH remainder exponentiates to $e_0$ (§The Kernel of the Exponential). For instance, with $a = e_1$ and $b = e_2$ the product $\exp(e_1)\exp(e_2)$ carries a term $\sin^2 1\, e_3$ that is absent from $\exp(e_1+e_2)$.

The Kernel of the Exponential

Because $\exp$ is surjective onto the group of units but not injective (§The Logarithm), the fibre over the identity measures the ambiguity of the logarithm.

Theorem. For a biquaternion $\tilde{Q}$ one has $\exp(\tilde{Q}) = e_0$ if and only if $\tilde{Q}$ is diagonalizable over $\mathbb{C}$ and every eigenvalue of $\tilde{Q}$ lies in $2\pi i\mathbb{Z}$.

Proof. If $\tilde{Q}$ is diagonalizable with eigenvalues $\lambda_1, \lambda_2$, then $\exp(\tilde{Q})$ is diagonalizable with eigenvalues $e^{\lambda_1}, e^{\lambda_2}$, and equals $I$ exactly when $e^{\lambda_1} = e^{\lambda_2} = 1$. Conversely, write $\tilde{Q} = S + N$ with $S$ semisimple and $N$ nilpotent commuting. Then $\exp(\tilde{Q}) = \exp(S)\exp(N)$ is the product of a semisimple and a unipotent factor; if it equals $I$, the unipotent factor is semisimple, hence trivial, so $N = 0$.

Concrete description. A biquaternion lies in the kernel exactly when its representing matrix is conjugate to $\mathrm{diag}(2\pi i m, 2\pi i n)$ with $m, n \in \mathbb{Z}$. The kernel is thus an infinite subset of $\mathbb{B}$, not an additive subgroup, since $\exp$ is not a homomorphism. It contains the scalars $2\pi i k\, e_0$ but much more: $\mathrm{diag}(0, 2\pi i)$ also exponentiates to the identity.

In biquaternion terms. The eigenvalues of the matrix representing $\tilde{Q}$ are $\lambda_\pm = Q_0 \pm iB$, with $B = \sqrt{Q_1^2 + Q_2^2 + Q_3^2}$. So $\exp(\tilde{Q}) = e_0$ exactly when either $\mathbf{Q} = 0$ and $\tilde{Q} = Q_0 e_0$ is a scalar with $Q_0 \in 2\pi i\mathbb{Z}$, or $B \neq 0$ and both $Q_0 \pm iB$ lie in $2\pi i\mathbb{Z}$. The second condition is equivalent to

$$ Q_0 = \pi i k, \qquad B = \pi j, \qquad k, j \in \mathbb{Z}, \quad k \equiv j \pmod 2, $$

the eigenvalues then being $2\pi i(k\pm j)/2$. If $B = 0$ but $\mathbf{Q} \neq 0$, the matrix is a nonzero nilpotent, hence not diagonalizable and not in the kernel: $\exp(\mathbf{Q}) = e_0 + \mathbf{Q} \neq e_0$.

Example. The biquaternion $\tilde{Q} = \pi i\, e_0 + \pi\, e_3$ has $Q_0 = \pi i$, $B = \pi$, so $k = j = 1$ and $\exp(\tilde{Q}) = e^{\pi i}(\cos\pi\, e_0 + \sin\pi\, e_3) = e_0$. Since $N(\tilde{Q}) = 0$, this kernel element is a zero divisor: the kernel is not contained in the group of units.

On the phrase "eigenvalues differ by a multiple of $2\pi i$". The condition implies such a difference, but the difference alone is weaker: $\tilde{Q} = e_0$ has equal eigenvalues yet $\exp(e_0) \neq e_0$. Each eigenvalue must be a multiple of $2\pi i$.

The Exponential in the Two Real Directions

Let $\hat{n}$ be a real unit vector part, so that $\hat{n}^2 = -e_0$. The two real three-dimensional families of exponents — the bivector directions $\theta\hat{n}$ and the vector directions $\psi\, i\hat{n}$, with $\theta, \psi \in \mathbb{R}$ — exponentiate in closed form. They are the two real summands of the trace-free subalgebra of $\mathbb{B}$, developed in Biquaternion Lie Algebra.

The bivector directions. For real $\theta$,

$$ \exp(\theta\hat{n}) = \cos\theta\, e_0 + \sin\theta\, \hat{n}, $$

a unit quaternion in $\mathbb{H}_{\mathbb{B}}$, since $N = \cos^2\theta + \sin^2\theta = 1$. The family is periodic with period $2\pi$, since $\exp(2\pi\hat{n}) = e_0$, and at the half-period $\exp(\pi\hat{n}) = -e_0$.

The vector directions. With $(i\hat{n})^2 = +e_0$ and real parameter $\psi \in \mathbb{R}$,

$$ \exp(\psi\, i\hat{n}) = \cosh\psi\, e_0 + \sinh\psi\, i\hat{n}. $$

This element is Hermitian, lies in $\mathbb{M}_+$, and has $N = \cosh^2\psi - \sinh^2\psi = 1$. Unlike the bivector family it is not periodic and is unbounded as $\psi \to \pm\infty$. The set of these elements is not a subgroup: the product of two of them in non-parallel directions is the product of an element of the same family and one of the bivector family.

Comparison. A general exponent is $\tilde{Q} = \mathbf{q} + i\mathbf{q}'$ with $\mathbf{q}, \mathbf{q}'$ real vector parts; the two families commute only when $\mathbf{q}$ and $\mathbf{q}'$ are parallel, since $[\mathbf{q}, i\mathbf{q}'] = 2i\,(\mathbf{q} \times \mathbf{q}')$, and in general the closed form of §The Exponential with $Q_0 = 0$ mixes the two families.

The Trigonometric and Hyperbolic Functions

Definitions

The trigonometric and hyperbolic functions are defined by the same power series as in the complex case:

$$ \sin(\tilde{Q}) = \sum_{n=0}^{\infty} \frac{(-1)^n \tilde{Q}^{2n+1}}{(2n+1)!}, \qquad \cos(\tilde{Q}) = \sum_{n=0}^{\infty} \frac{(-1)^n \tilde{Q}^{2n}}{(2n)!}, $$

$$ \sinh(\tilde{Q}) = \sum_{n=0}^{\infty} \frac{\tilde{Q}^{2n+1}}{(2n+1)!}, \qquad \cosh(\tilde{Q}) = \sum_{n=0}^{\infty} \frac{\tilde{Q}^{2n}}{(2n)!}. $$

These series converge for every $\tilde{Q} \in \mathbb{B}$, for the same reason as the exponential.

Computation via the Exponential

The trigonometric and hyperbolic functions can be expressed in terms of the exponential:

$$ \sin(\tilde{Q}) = \frac{\exp(i\tilde{Q}) - \exp(-i\tilde{Q})}{2i}, \qquad \cos(\tilde{Q}) = \frac{\exp(i\tilde{Q}) + \exp(-i\tilde{Q})}{2}, $$

$$ \sinh(\tilde{Q}) = \frac{\exp(\tilde{Q}) - \exp(-\tilde{Q})}{2}, \qquad \cosh(\tilde{Q}) = \frac{\exp(\tilde{Q}) + \exp(-\tilde{Q})}{2}, $$

where $i$ is the scalar imaginary. These identities hold because $i$ commutes with everything, so the standard derivations carry over. They reduce the computation to the exponential formula.

The Case $B \neq 0$

For $\tilde{Q} = Q_0 e_0 + \mathbf{Q}$ with $B \neq 0$ and $\hat{n} = \mathbf{Q}/B$, we have

$$ \exp(\tilde{Q}) = e^{Q_0}(\cos B \, e_0 + \sin B \, \hat{n}), \qquad \exp(-\tilde{Q}) = e^{-Q_0}(\cos B \, e_0 - \sin B \, \hat{n}). $$

Substituting into the formulas for the trigonometric and hyperbolic functions, we obtain

$$ \sin(\tilde{Q}) = \sin(Q_0) \cosh B \, e_0 + \cos(Q_0) \sinh B \, \hat{n}, $$

$$ \cos(\tilde{Q}) = \cos(Q_0) \cosh B \, e_0 - \sin(Q_0) \sinh B \, \hat{n}, $$

$$ \sinh(\tilde{Q}) = \sinh(Q_0) \cos B \, e_0 + \cosh(Q_0) \sin B \, \hat{n}, $$

$$ \cosh(\tilde{Q}) = \cosh(Q_0) \cos B \, e_0 + \sinh(Q_0) \sin B \, \hat{n}. $$

The Case $B = 0$

If $B = 0$, then $\mathbf{Q}^2 = 0$, and the powers of $\tilde{Q} = Q_0 e_0 + \mathbf{Q}$ satisfy

$$ \tilde{Q}^n = Q_0^n e_0 + n Q_0^{n-1} \mathbf{Q}, \qquad n \geq 1, $$

and $\tilde{Q}^0 = e_0$. This follows from the binomial expansion, which truncates at first order in $\mathbf{Q}$ because $\mathbf{Q}^2 = 0$. Substituting into the power series of the trigonometric and hyperbolic functions, and using the Taylor series of the complex sine and cosine, we obtain

$$ \sin(\tilde{Q}) = \sin(Q_0) e_0 + \cos(Q_0) \mathbf{Q}, $$

$$ \cos(\tilde{Q}) = \cos(Q_0) e_0 - \sin(Q_0) \mathbf{Q}, $$

$$ \sinh(\tilde{Q}) = \sinh(Q_0) e_0 + \cosh(Q_0) \mathbf{Q}, $$

$$ \cosh(\tilde{Q}) = \cosh(Q_0) e_0 + \sinh(Q_0) \mathbf{Q}. $$

The Case $\mathbf{Q} = 0$

If $\mathbf{Q} = 0$, both formulas reduce to the ordinary complex trigonometric and hyperbolic functions.

Summary of Trigonometric and Hyperbolic Functions

$$ \sin(\tilde{Q}) = \begin{cases} \sin(Q_0) \cosh B \, e_0 + \cos(Q_0) \sinh B \, \hat{n} & \text{if } B \neq 0, \\[1mm] \sin(Q_0) e_0 + \cos(Q_0) \mathbf{Q} & \text{if } B = 0, \end{cases} $$

$$ \cos(\tilde{Q}) = \begin{cases} \cos(Q_0) \cosh B \, e_0 - \sin(Q_0) \sinh B \, \hat{n} & \text{if } B \neq 0, \\[1mm] \cos(Q_0) e_0 - \sin(Q_0) \mathbf{Q} & \text{if } B = 0, \end{cases} $$

$$ \sinh(\tilde{Q}) = \begin{cases} \sinh(Q_0) \cos B \, e_0 + \cosh(Q_0) \sin B \, \hat{n} & \text{if } B \neq 0, \\[1mm] \sinh(Q_0) e_0 + \cosh(Q_0) \mathbf{Q} & \text{if } B = 0, \end{cases} $$

$$ \cosh(\tilde{Q}) = \begin{cases} \cosh(Q_0) \cos B \, e_0 + \sinh(Q_0) \sin B \, \hat{n} & \text{if } B \neq 0, \\[1mm] \cosh(Q_0) e_0 + \sinh(Q_0) \mathbf{Q} & \text{if } B = 0. \end{cases} $$

Properties

Pythagorean identity. The identity

$$ \sin^2(\tilde{Q}) + \cos^2(\tilde{Q}) = e_0 $$

holds for every $\tilde{Q} \in \mathbb{B}$.

Proof. Write $A = e^{i\tilde{Q}}$ and $B = e^{-i\tilde{Q}}$. Since $i\tilde{Q}$ commutes with $-i\tilde{Q}$, we have $AB = e^{i\tilde{Q}}e^{-i\tilde{Q}} = e^{i\tilde{Q}-i\tilde{Q}} = e_0$. Then

$$ \sin^2(\tilde{Q}) + \cos^2(\tilde{Q}) = \frac{(A - B)^2}{-4} + \frac{(A + B)^2}{4} = \frac{-(A^2 - 2AB + B^2) + (A^2 + 2AB + B^2)}{4} = \frac{4AB}{4} = AB = e_0. $$

Alternatively, in the oscillatory regime, a direct expansion gives

$$ \sin^2(\tilde{Q}) + \cos^2(\tilde{Q}) = (\cosh^2 B - \sinh^2 B) e_0 = e_0, $$

since the vector parts cancel by antisymmetry and the scalar parts combine via the hyperbolic identity.

The reason the identity survives the non-commutativity of $\mathbb{B}$ is that both $\sin(\tilde{Q})$ and $\cos(\tilde{Q})$ lie in the commutative subalgebra $\mathbb{C}[\tilde{Q}]$ generated by $\tilde{Q}$, within which the standard complex-analytic identities hold. The identity fails only for expressions involving two or more independent biquaternions (see the addition formulas below).

Hyperbolic identity. The identity

$$ \cosh^2(\tilde{Q}) - \sinh^2(\tilde{Q}) = e_0 $$

holds for every $\tilde{Q} \in \mathbb{B}$.

Proof. Write $A = e^{\tilde{Q}}$ and $B = e^{-\tilde{Q}}$. Since $\tilde{Q}$ commutes with $-\tilde{Q}$, we have $AB = e^{\tilde{Q}-\tilde{Q}} = e_0$. Then

$$ \cosh^2(\tilde{Q}) - \sinh^2(\tilde{Q}) = \frac{(A + B)^2 - (A - B)^2}{4} = \frac{(A^2 + 2AB + B^2) - (A^2 - 2AB + B^2)}{4} = \frac{4AB}{4} = AB = e_0. $$

Double-angle formulas. The standard double-angle formulas hold:

$$ \sin(2\tilde{Q}) = 2\sin(\tilde{Q})\cos(\tilde{Q}), \qquad \cos(2\tilde{Q}) = \cos^2(\tilde{Q}) - \sin^2(\tilde{Q}), $$

$$ \sinh(2\tilde{Q}) = 2\sinh(\tilde{Q})\cosh(\tilde{Q}), \qquad \cosh(2\tilde{Q}) = \cosh^2(\tilde{Q}) + \sinh^2(\tilde{Q}). $$

These follow from the corresponding identities in the commutative subalgebra $\mathbb{C}[\tilde{Q}]$.

Addition formulas. The addition formulas, such as

$$ \sin(\tilde{P} + \tilde{Q}) = \sin(\tilde{P})\cos(\tilde{Q}) + \cos(\tilde{P})\sin(\tilde{Q}), $$

hold if $\tilde{P}$ and $\tilde{Q}$ commute. In general they fail. For instance, with $\tilde{P} = e_1$ and $\tilde{Q} = e_2$, each is pure with complex norm $1$, so

$$ \sin(e_1) = \sinh(1) e_1, \qquad \cos(e_1) = \cosh(1) e_0, $$

$$ \sin(e_2) = \sinh(1) e_2, \qquad \cos(e_2) = \cosh(1) e_0, $$

and hence

$$ \sin(e_1)\cos(e_2) + \cos(e_1)\sin(e_2) = \sinh(1)\cosh(1)(e_1 + e_2) = \tfrac{1}{2}\sinh(2)(e_1 + e_2). $$

On the other hand, $e_1 + e_2$ is pure with complex norm $\sqrt{2}$, so

$$ \sin(e_1 + e_2) = \frac{\sinh\sqrt{2}}{\sqrt{2}}(e_1 + e_2). $$

These are unequal: $\tfrac{1}{2}\sinh(2) \approx 1.813$, whereas $\sinh\sqrt{2}/\sqrt{2} \approx 1.368$. The failure is a genuine consequence of non-commutativity.

Relation to the exponential. The relations $\sin(\tilde{Q}) = (\exp(i\tilde{Q}) - \exp(-i\tilde{Q}))/(2i)$ and so on hold in general, because the scalar imaginary $i$ commutes with everything.

The Logarithm

Definition

The logarithm of a biquaternion $\tilde{Q}$ is defined as the inverse of the exponential:

$$ \log(\tilde{Q}) = \tilde{L} \iff \exp(\tilde{L}) = \tilde{Q}. $$

The exponential is never a zero divisor: from the property $N(\exp(\tilde{L})) = e^{2 L_0}$ for every $\tilde{L}$, the biquaternion norm of $\exp(\tilde{L})$ is always nonzero. So the logarithm can exist only on the group of units $\mathbb{B}^\times = \{N(\tilde{Q}) \neq 0\}$. The following theorem shows that it exists on the whole group of units, and on no larger set.

Existence of the Logarithm

Theorem. Every invertible biquaternion has a logarithm. The domain of the logarithm is exactly the group of units

$$ \mathbb{B}^\times = \{\tilde{Q} \in \mathbb{B} : N(\tilde{Q}) \neq 0\}. $$

Proof. Let $\tilde{Q}$ be invertible, so $N(\tilde{Q}) \neq 0$. We treat the three cases.

Case $B \neq 0$. Set $R = \sqrt{N(\tilde{Q})}$, a fixed square root, $\hat{n} = \mathbf{Q}/B$, and let $\Theta$ be determined by $\cos\Theta = Q_0/R$ and $\sin\Theta = B/R$, so that $Q_0 = R\cos\Theta$ and $B = R\sin\Theta$. Since $N(\tilde{Q}) \neq 0$, we have $R \neq 0$, so the principal branch of $\log R$ is defined. Set

$$ \tilde{L} = \log R \, e_0 + \Theta \hat{n}. $$

Then

$$ \exp(\tilde{L}) = R(\cos\Theta \, e_0 + \sin\Theta \, \hat{n}) = Q_0 e_0 + B \hat{n} = \tilde{Q}, $$

using the identity $\exp(\Theta \hat{n}) = \cos\Theta \, e_0 + \sin\Theta \, \hat{n}$, which follows from $\hat{n}^2 = -e_0$.

Case $B = 0$, $\mathbf{Q} \neq 0$. Then $\mathbf{Q}^2 = 0$ and $N(\tilde{Q}) = Q_0^2 \neq 0$, so $Q_0 \neq 0$. Set

$$ \tilde{L} = \log Q_0 \, e_0 + \mathbf{Q}/Q_0. $$

Then

$$ \tilde{L}^2 = (\log Q_0)^2 e_0 + 2 \log Q_0 \cdot \mathbf{Q}/Q_0 + (\mathbf{Q}/Q_0)^2. $$

The last term vanishes because $\mathbf{Q}^2 = 0$. The middle term commutes with the first term, since it is a scalar multiple of $\mathbf{Q}$ and the first term is a scalar multiple of $e_0$. So

$$ \exp(\tilde{L}) = \exp(\log Q_0 \cdot e_0) \exp(\mathbf{Q}/Q_0) = Q_0 (e_0 + \mathbf{Q}/Q_0) = Q_0 e_0 + \mathbf{Q} = \tilde{Q}. $$

Case $\mathbf{Q} = 0$. Then $\tilde{Q} = Q_0 e_0$ with $Q_0 \neq 0$ (since $N(\tilde{Q}) = Q_0^2 \neq 0$), and $\log \tilde{Q} = \log Q_0 \cdot e_0$, the ordinary complex logarithm.

Converse. If $\tilde{Q}$ has a logarithm, then $\tilde{Q} = \exp(\tilde{L})$ for some $\tilde{L}$, and $N(\tilde{Q}) = N(\exp(\tilde{L})) = e^{2 L_0} \neq 0$. So $\tilde{Q}$ is invertible.

Summary of the Logarithm

$$ \log(\tilde{Q}) = \begin{cases} \log R \, e_0 + \Theta \hat{n} & \text{if } B \neq 0, \\[2mm] \log Q_0 \, e_0 + \dfrac{\mathbf{Q}}{Q_0} & \text{if } B = 0, \; \mathbf{Q} \neq 0, \\[2mm] \log Q_0 \, e_0 & \text{if } \mathbf{Q} = 0, \end{cases} $$

defined on the group of units $\mathbb{B}^\times = \{N(\tilde{Q}) \neq 0\}$. Here $R = \sqrt{Q_0^2 + B^2}$, $\cos\Theta = Q_0/R$, and $\sin\Theta = B/R$. The branch is determined by the choice of $R$, the choice of $\Theta$, and the choice of branch of the complex logarithm.

Properties

Multiplicativity. The logarithm satisfies $\log(\tilde{P} \tilde{Q}) = \log(\tilde{P}) + \log(\tilde{Q})$ whenever $\tilde{P}$ and $\tilde{Q}$ commute. In general the logarithm is not multiplicative, and the converse is not claimed: as for the exponential group law (§The Group Law: $\exp(a)\exp(b)$ versus $\exp(a+b)$), a non-commuting pair may satisfy the identity under a compatible choice of branches. The failure is the same as in the quaternion case: when the two axes $\hat{n}$ and $\hat{m}$ do not commute, the product of the exponentials is not the exponential of the sum.

Multivaluedness. The logarithm is multivalued. The branches are parameterized by the integer shifts of the scalar logarithm and of the angle $\Theta$, that is, by $\log R \mapsto \log R + 2\pi i n$ and $\Theta \mapsto \Theta + 2\pi m$ with $n, m \in \mathbb{Z}$; equivalently, they differ by the elements of the kernel of the exponential.

Domain. The logarithm exists exactly on the group of units $\mathbb{B}^\times$. It fails on the zero divisor set $\mathcal{Z}$ and at the zero element $0$.

Failure at pure nilpotents. In particular, the logarithm does not exist at any pure nilpotent $\tilde{Q} = \mathbf{Q}$ with $\mathbf{Q}^2 = 0$ and $\mathbf{Q} \neq 0$: such an element is a nonzero zero divisor, hence not invertible, whereas every exponential is invertible.

The Power Functions

Definition

For $\tilde{Q} \in \mathbb{B}$ and $\alpha \in \mathbb{C}$, the power function is defined by

$$ \tilde{Q}^\alpha = \exp(\alpha \log \tilde{Q}). $$

The power function inherits the multivaluedness of the logarithm: for non-integer $\alpha$, it is multivalued, and the principal branch is obtained from the principal logarithm. The domain is the group of units $\mathbb{B}^\times$.

The Case $B \neq 0$

Using the formula for the logarithm and the exponential, we obtain

$$ \tilde{Q}^\alpha = R^\alpha \bigl(\cos(\alpha\Theta) \, e_0 + \sin(\alpha\Theta) \, \hat{n}\bigr), \qquad B \neq 0, $$

where $R = \sqrt{Q_0^2 + B^2}$, $\cos\Theta = Q_0/R$, and $\sin\Theta = B/R$.

The Case $B = 0$, $\mathbf{Q} \neq 0$

In the nilpotent regime,

$$ \tilde{Q}^\alpha = Q_0^\alpha e_0 + \alpha Q_0^{\alpha-1} \mathbf{Q}, \qquad B = 0, \quad \mathbf{Q} \neq 0, $$

with $Q_0 \neq 0$ required for invertibility. This involves the linear term $\alpha Q_0^{\alpha-1} \mathbf{Q}$, which is the analogue of the expansion $(1 + a)^\alpha \approx 1 + \alpha a$ for small $a$ with $a^2 = 0$.

The Case $\mathbf{Q} = 0$

If $\mathbf{Q} = 0$, the power function reduces to the ordinary complex power.

Summary of the Power Function

$$ \tilde{Q}^\alpha = \begin{cases} R^\alpha \bigl(\cos(\alpha\Theta) \, e_0 + \sin(\alpha\Theta) \, \hat{n}\bigr) & \text{if } B \neq 0, \\[2mm] Q_0^\alpha e_0 + \alpha Q_0^{\alpha-1} \mathbf{Q} & \text{if } B = 0, \; \mathbf{Q} \neq 0, \end{cases} $$

defined on $\mathbb{B}^\times$.

Properties

Integer powers. For integer $n$, the power function is single-valued on $\mathbb{B}^\times$ and reduces to the ordinary power $\tilde{Q}^n$. Moreover, for integer $n$ the ordinary power $\tilde{Q}^n$ is defined by repeated multiplication on all of $\mathbb{B}$, including the zero divisors; the formula $\tilde{Q}^n = \exp(n \log \tilde{Q})$ holds only on the group of units.

Non-multiplicativity. In general, $(\tilde{P} \tilde{Q})^\alpha \neq \tilde{P}^\alpha \tilde{Q}^\alpha$, because the logarithm is not multiplicative.

The square root. The square root $\tilde{Q}^{1/2}$ is multivalued, and the two branches correspond to the two signs of the angle.

Consistency with the biquaternion norm. For a fixed choice of branch, the power function satisfies

$$ N(\tilde{Q}^\alpha) = N(\tilde{Q})^\alpha. $$

Proof. In the oscillatory regime, $\tilde{Q}^\alpha = R^\alpha(\cos(\alpha\Theta) e_0 + \sin(\alpha\Theta) \hat{n})$, so

$$ N(\tilde{Q}^\alpha) = R^{2\alpha}(\cos^2(\alpha\Theta) + \sin^2(\alpha\Theta)) = R^{2\alpha} = (R^2)^\alpha = N(\tilde{Q})^\alpha. $$

In the nilpotent regime, $\tilde{Q}^\alpha = Q_0^\alpha e_0 + \alpha Q_0^{\alpha-1}\mathbf{Q}$, so

$$ N(\tilde{Q}^\alpha) = (Q_0^\alpha)^2 + 0 = Q_0^{2\alpha} = N(\tilde{Q})^\alpha. $$

The identity is branch-dependent: a consistent choice of branch must be made for $R^\alpha$ and $R^{2\alpha}$ (respectively $Q_0^\alpha$ and $Q_0^{2\alpha}$), since the multivaluedness of the power means that the two sides can differ by a phase if inconsistent branches are chosen. Equivalently, one must use a compatible choice of the complex logarithm on both sides of the identity.

Relations to the Quaternion and Complex Cases

The Quaternion Case

The biquaternion algebra contains the quaternion algebra $\mathbb{H}$ as the subspace $\mathbb{H}_{\mathbb{B}}$, which is the fixed-point set of complex conjugation. For a quaternion $q = q_0 + \mathbf{q}$ with real coefficients, the complex norm $B = |\mathbf{q}|$ is real and non-negative, and the elementary functions reduce to the ordinary quaternion elementary functions:

$$ \exp(q) = e^{q_0}(\cos|\mathbf{q}| + \sin|\mathbf{q}| \, \hat{n}), $$

$$ \log(q) = \log|q| + \arccos(q_0/|q|) \hat{n}, $$

and so on, where $|q| = \sqrt{q_0^2 + |\mathbf{q}|^2}$ is the quaternion norm. These are the standard formulas of quaternion analysis.

The Complex Case

The biquaternion algebra contains the complex numbers as the subspace $\mathbb{C}_{\mathbb{B}}$, which is the fixed-point set of quaternion conjugation (the scalar part). For a complex scalar $\tilde{Q} = Q_0 e_0$, the vector part vanishes, and the elementary functions reduce to the ordinary complex elementary functions:

$$ \exp(Q_0 e_0) = e^{Q_0} e_0, \qquad \log(Q_0 e_0) = \log(Q_0) e_0, $$

and so on. These are the standard formulas of complex analysis.

The Relation Between the Two

The quaternion case and the complex case are the two extremes of the biquaternion case: the quaternion case is the case where $B$ is real, and the complex case is the case where $\mathbf{Q} = 0$. The general biquaternion case interpolates between the two, with $B$ complex and $\mathbf{Q}$ nonzero.

Non-Commutativity and the One-Variable Case

The elementary functions of a biquaternion variable are as simple as they are because the vector part is a single element, and it commutes with itself. Equivalently, all elementary functions of a single biquaternion $\tilde{Q}$ lie in the commutative subalgebra $\mathbb{C}[\tilde{Q}]$ generated by $\tilde{Q}$ and $e_0$. In the oscillatory regime or the nonzero nilpotent regime, this subalgebra is exactly two-dimensional over $\mathbb{C}$, and it is isomorphic either to $\mathbb{C} \times \mathbb{C}$ (in the oscillatory regime) or to $\mathbb{C}[\epsilon]/(\epsilon^2)$ (in the nilpotent regime). Within either isomorphism type, the standard complex-analytic identities — including the Pythagorean identity, the hyperbolic identity, the double-angle formulas, and the relation between the exponential and the trigonometric functions — hold, because the algebra is commutative.

The powers of $\mathbf{Q}$ are determined by the single relation $\mathbf{Q}^2 = -B^2 e_0$ (in the oscillatory regime) or $\mathbf{Q}^2 = 0$ (in the nilpotent regime), and the whole power series can be summed in closed form. The two cases correspond to the two isomorphism types of $\mathbb{C}[\tilde{Q}]$.

For functions of two or more biquaternion variables, the situation is different. The powers of a sum $\tilde{P} + \tilde{Q}$ involve the products $\tilde{P} \tilde{Q}$ and $\tilde{Q} \tilde{P}$, which are not equal in general. The binomial expansion does not hold, and the exponential of a sum is not in general the product of the exponentials when the two biquaternions do not commute. So the elementary functions of several biquaternion variables are much more complicated than the elementary functions of one variable, and their theory is largely open.

This is the fundamental reason the theory of the elementary functions of a biquaternion variable is tractable, while the theory of functions of several biquaternion variables is not.

Collected Formulas

In the table, $B = \sqrt{Q_1^2 + Q_2^2 + Q_3^2}$, $\hat{n} = \mathbf{Q}/B$, $R = \sqrt{Q_0^2 + B^2}$, $\cos\Theta = Q_0/R$, and $\sin\Theta = B/R$. The exponential, the trigonometric functions, and the hyperbolic functions are defined for every $\tilde{Q} \in \mathbb{B}$. The logarithm and the power function are defined only on the group of units $N(\tilde{Q}) \neq 0$; in the second column, the formulas for the logarithm and the power function assume $Q_0 \neq 0$, which is automatically implied by $N(\tilde{Q}) \neq 0$ when $B = 0$.

Open Questions

1. Functions of several biquaternion variables. How do the elementary functions extend to functions of two or more biquaternion variables? The non-commutativity is a serious obstruction, and the theory is largely open.

2. The transition between the two regimes. As $B \to 0$, the oscillatory formulas degenerate into the nilpotent ones. What is the precise nature of this degeneration? Is there a uniform asymptotic expansion that interpolates between the two regimes?

3. The complex norm. What is the geometric or algebraic meaning of the complex norm $B$ when it is not real? The complex norm is a square root of the biquaternion norm, and it is complex for a general biquaternion, but its interpretation is not clear. In particular, $B$ is the analogue of the "magnitude" of the vector part, but its complex-valuedness means it has both a modulus and a phase.

4. The relation to the analysis. How do the elementary functions interact with the differential operators of the analysis article? For example, what is $\tilde{\nabla} \exp(\tilde{Q})$ for a general biquaternion $\tilde{Q}$?

5. The logarithm at the boundary of the group of units. The logarithm exists on $\mathbb{B}^\times$ but not on its boundary, which is the zero divisor set $\mathcal{Z}$ together with $0$. What is the correct object that replaces the logarithm on the boundary? The answer is likely to involve a formal logarithm or a deformation of the algebra.

6. The derivative of the exponential in the vector directions. The formula $\partial_k \exp(\tilde{Q}) = \int_0^1 e^{s \tilde{Q}} e_k e^{(1-s)\tilde{Q}}\,ds$ is the biquaternion analogue of the usual identity for the derivative of the exponential. What are the consequences of this formula for the analysis of biquaternion-valued functions?

Summary

Every elementary function is defined by applying the scalar power series to $\tilde{Q}$, and all its values lie in the commutative subalgebra $\mathbb{C}[\tilde{Q}]$ generated by $\tilde{Q}$. This is why the standard identities of complex analysis — the Pythagorean identity, the hyperbolic identity, the double-angle formulas, the relation between the exponential and the trigonometric functions — survive the non-commutativity of $\mathbb{B}$ for a single variable; the addition formulas for two independent biquaternions do not, and fail for explicit counterexamples such as $\tilde{P} = e_1$, $\tilde{Q} = e_2$.

The structure of $\mathbb{C}[\tilde{Q}]$ is governed by the discriminant and yields exactly two regimes. In the oscillatory regime $B \neq 0$ the subalgebra is $\mathbb{C} \times \mathbb{C}$, the element $\tilde{Q}$ is diagonalizable with eigenvalues $Q_0 \pm iB$, and the closed forms are the oscillatory-hyperbolic expressions of the collected table, built on $\tilde{Q} = Q_0 e_0 + B\hat{n}$ with $\hat{n}^2 = -e_0$. In the nilpotent regime $B = 0$, $\mathbf{Q} \neq 0$, the subalgebra is $\mathbb{C}[\epsilon]/(\epsilon^2)$ with the nilpotent $\epsilon = \mathbf{Q}$, and every power series truncates after the linear term: the exponential becomes $e^{Q_0}(e_0 + \mathbf{Q})$, the logarithm becomes $\log Q_0\,e_0 + \mathbf{Q}/Q_0$, and the higher functions reduce to their first-order Taylor expansions in $\mathbf{Q}$.

The exponential is the model of the theory. It is never zero, it satisfies $\exp(\tilde{P}+\tilde{Q}) = \exp(\tilde{P})\exp(\tilde{Q})$ when $\tilde{P}$ and $\tilde{Q}$ commute (the converse can fail, §The Group Law: $\exp(a)\exp(b)$ versus $\exp(a+b)$), and the logarithm inverts it on the group of units, using the decomposition $\tilde{Q} = R(\cos\Theta\,e_0 + \sin\Theta\,\hat{n})$ with $R = \sqrt{N(\tilde{Q})}$, $\cos\Theta = Q_0/R$ and $\sin\Theta = B/R$, from which the power function $\tilde{Q}^{\alpha}$ is obtained. Because the scalar imaginary $i$ commutes with everything, it relates the trigonometric functions to the exponential and the trigonometric to the hyperbolic functions, but only for a single argument in a common plane.

The exponential, the trigonometric and the hyperbolic functions are defined for every $\tilde{Q} \in \mathbb{B}$; the logarithm and the power function are defined only on the group of units $N(\tilde{Q}) \neq 0$, and the logarithm there is many-valued, the ambiguity coming from the multi-valuedness of $\Theta$ and of the scalar logarithm.

Summary of Notation

Further Reading

• William Rowan Hamilton, Lectures on Quaternions (Hodges and Smith, Dublin, 1853), for the original formulation of the quaternion exponential and logarithm.
• J. P. Ward, Quaternions and Cayley Numbers: Algebra and Applications (Kluwer, Dordrecht, 1997), Chapter 3, for the elementary functions of biquaternions.
• S. J. Sangwine, T. A. Ell, and N. Le Bihan, "Fundamental representations and algebraic properties of biquaternions or complexified quaternions", Advances in Applied Clifford Algebras 21 (2011) 607–636, for the exponential of complexified quaternions.
• S. J. Sangwine, "Biquaternion (complexified quaternion) roots of $-1$", Advances in Applied Clifford Algebras 16 (2006) 63–68, for the classification of the roots of $-1$ that serve as axes of the vector part.
• S. J. Sangwine and D. Alfsmann, "Determination of the biquaternion divisors of zero, including the idempotents and nilpotents", Advances in Applied Clifford Algebras 20 (2010) 401–416, for the zero divisors and nilpotents.
• Pertti Lounesto, Clifford Algebras and Spinors (Cambridge, 2001), for the connection to Clifford algebras.