Quaternionic Structures in Mathematics and Physics SISSA - Trieste, 5-9 September 1994

QUATERNIONIC DERIVABILITY II

by Sorin Lugojan

Technical University of Timisoara Mechanics Faculty-Mathematical Department bd. Mihai Viteazu nr 1, 1900-Timisoara Romania

AMS subject classification: 30G99

Abstract.

The article proposes a new type of complex and quaternionic derivative.

Since the hard quaternion derivatives:

f′_{r}(w_{0}) = lim_{{w→w_{0} \atop w≠w_{0}}} \{[f(w) − f(w_{0})] (w − w_{0})^{−1}\}  (1)
f′_{l}(w_{0}) = lim_{{w→w_{0} \atop w≠w_{0}}} \{(w − w_{0})^{−1} [f(w) − f(w_{0})]\}

give reduced sets of right or left quaternionic derivable functions (see [1], [2], [3], [7], [10]) the study of softer quaternionic derivatives appeared (see [1], [4], [5], [6], [8], [10]). The present article tries to relaunch (for the sake of physics applications) the study of the hard form of the quaternionic derivative in a generalized manner.

1. Preliminaries.

1.1. Definition. We define: 1.) H_{1} = {w ∈ H | w = ix_{2} + jx_{3} + kx_{4}}, analogously for H_{2}, H_{3}, H_{4}. 2.) H_{1,2} = {w ∈ H | w = jx_{3} + kx_{4}}, analogously for H_{1,3}, …, H_{3,4}. 3.) H_{1,2,3} = {w ∈ H | w = kx_{4}}, analogously for H_{1,2,4}, H_{1,3,4}, H_{2,3,4}.

One can raise now the following question: are there functions, derivable in the sense of the formula (1), between sets belonging to the Definition 1.1.? The answer is given by the following:

1.2. Theorem 1.) The only functions f : Q ⊆ H_{i} ⟶ H (∀i ∈ {1, 2, 3, 4} and Q an open set of the induced topology on H_{i}) which have the property that f′(w_{0}) = lim_{{w→w_{0} \atop w≠w_{0}}} \{[f(w) − f(w_{0})] (w − w_{0})^{−1}\} exists and is finite, ∀w_{0} ∈ Q, are the constant

functions. 2.) The only functions f : Q ⊆ H ⟶ H_{i} (∀i ∈ {1, 2, 3, 4} and Q an open set) which have the property that f′(w_{0}) exists and is finite, ∀w_{0} ∈ Q, are the constant functions.

3.) The only functions f : Q ⊆ H_{i} ⟶ H_{j} (∀i, j ∈ {1, 2, 3, 4} and Q an open set of the induced topology) which have the property that f′(w_{0}) exists and is finite, ∀w_{0} ∈ Q, are the constant functions. 4.) The only functions f : Q ⊆ H_{i,j} ⟶ H (∀i ≠ j, i, j ∈ {1, 2, 3, 4} and Q an open set of the induced topology) which have the property that f′(w_{0}) exists and is finite, ∀w_{0} ∈ Q, are the constant functions. 5.) The only functions f : Q ⊆ H ⟶ H_{i,j} (∀i = j, i, j ∈ {1, 2, 3, 4} and Q an open set) which have the property that f′(w_{0}) exists and is finite, ∀w_{0} ∈ Q, are the constant functions. 6.) For the functions f : Q ⊆ H_{i,j} ⟶ H_{k} (∀i ≠ j, i, j, k ∈ {1, 2, 3, 4} and Q an open set in the induced topology) which have the property that f′(w_{0}) exists and is finite, ∀w_{0} ∈ Q, one obtains the following results: a. if i = 3, j = 4, k = 4, then f has the form:

f(x_{1} + ix_{2}) = h(x_{1} + ix_{2}) + jC, where h : Q ⊆ C ⟶ C is an holomorph function, b. if i = 3, j = 4, k = 3, then f(x_{1} + ix_{2}) = h(x_{1} + ix_{2}) + kC, c. if i = 3, j = 4, k = 2, then f(x_{1} + ix_{2}) = c + jh(x_{2} + ix_{1}). Analogously for the rest of the cases. 7.) The only functions f : Q ⊆ H_{k} ⟶ H_{i,j} (∀i ≠ j, i, j, k ∈ {1, 2, 3, 4} and Q an open set of the induced topology) which have the property that f′(w_{0}) exists and is finite, ∀w_{0} ∈ Q, are the constant functions. 8.) For the functions f : Q ⊆ H_{i,j} ⟶ H_{k,l} (∀i, j, k, l ∈ {1, 2, 3, 4} i ≠ j, k ≠ l and Q an open set of the induced topology) which have the property that f′(w_{0}) exists and is finite, ∀w_{0} ∈ Q, one can say that: a. they are constant functions if {i, j} ∩ {k, l} has exact one element, b. they are equivalent with the holomorph functions if {i, j} = {k, l} or {i, j} ∩ {k, l} = ∅. 9.) The functions f : Q ⊆ H_{i,j,k} ⟶ H (∀i, j, k ∈ {1, 2, 3, 4} i ≠ j, i ≠ k, j ≠ k and Q an open set of the induced topology) which have the property that f′(w_{0}) exists and is finite, ∀w_{0} ∈ Q, are equivalent with the derivable functions f : Q ⊆ IR ⟶ IR^{4}. 10.) The only functions f : Q ⊆ H ⟶ H_{i,j,k} (∀i, j, k ∈ {1, 2, 3, 4} i ≠ j, i ≠ k, j ≠ k and Q an open set) which have the property that f′(w_{0}) exists and is finite, ∀w_{0} ∈ Q, are the constant functions.

Proof. The extended proof is analogous to that of the Theorem from [3] and has around 30 pages. As an example, we prove 6.) c.

Let Q be an open set from H_{3,4}, w_{0} ∈ Q and f : Q ⟶ H_{2}, hence w = x_{1} + ix_{2}, f = f(w) and f = f_{1} + jf_{2} + kf_{3}. According to the notations of [3] we have:

w̄ = x_{1} − ix_{2} , x_{1} = (1/2) (w + w̄) , x_{2} = −(i/2) (w − w̄).

Therefore

df = (∂f/∂x_{1}) dx_{1} + (∂f/∂x_{2}) dx_{2} =
= (∂f/∂x_{1}) (1/2) (dw + dw̄) − (∂f/∂x_{2}) (i/2)(dw − dw̄) =
= (1/2) [(∂f/∂x_{1}) − (∂f/∂x_{2}) i] dw + (1/2) [(∂f/∂x_{1}) + (∂f/∂x_{2}) i] dw̄.

Following an analogous reasoning of that from the proof of the Theorem just mentioned above, it follows that the limit

lim_{{w→w_{0} \atop w≠w_{0}}} \{[f(w) − f(w_{0})] (w − w_{0})^{−1}\}

exsists and is finite iff:

∂f/∂x_{1} + (∂f/∂x_{2}) i = 0 and this is equivalent to
∂f_{1}/∂x_{1} + j∂f_{2}/∂x_{1} + k∂f_{3}/∂x_{1} + (∂f_{1}/∂x_{2} + j∂f_{2}/∂x_{2} + k∂f_{3}/∂x_{2}) i = 0

which is equivalent to

\{ \begin{matrix}∂f_{1}/∂x_{1} = 0 \\ ∂f_{1}/∂x_{2} = 0 \\ ∂f_{2}/∂x_{2} = ∂f_{3}/∂x_{1} \\ ∂f_{2}/∂x_{2} = −∂f_{3}/∂x_{2} & q.e.d. \end{matrix}

One obtains analogous results if there are analyzed the following types of H-derivability definitions:

lim_{{w→w_{0} \atop w≠w_{0}}} \{[f(w) − f(w_{0})] (w̄ − w̄_{0})^{−1}\}

lim_{{w→w_{0} \atop w≠w_{0}}} \{[f(w) − f(w_{0})] (w̃ − w̃_{0})^{−1}\}

lim_{{w→w_{0} \atop w≠w_{0}}} \{[f(w) − f(w_{0})] (ŵ − ŵ_{0})^{−1}\}.

(We recall some notations of the [3]:

\begin{matrix}w = x_{1} + ix_{2} + jx_{3} + kx_{4}, & w̄ = x_{1} − ix_{2} − jx_{3} − kx_{4}, \\ w̃ = −x_{1} + ix_{2} − jx_{3} − kx_{4}, & ŵ = −x_{1} − ix_{2} + jx_{3} − kx_{4}.) \end{matrix}

So, it is obvious that if someone wants to use a hard H-derivative, it is necessary an extension of the notion of derivative. It is just what we are doing in the followings.

2. An extension of the C-derivative

First we present the idea of the H-derivative for the particular case of the complex numbers.

Throughout this paragraph Q is an open set of C, z_{0} ∈ Q, z = x + iy and f : Q ⊆ C ⟶ C such that f ∈ C^{1} (Q ⊆ IR^{2}, IR^{2}). For f we have the decomposition:

df = (∂f/∂x) dx + (∂f/∂y) dy =
= (∂f/∂x) (1/2)(dz + dz̄) − (∂f/∂y) (i/2)(dz − dz̄) =
= (1/2) (∂f/∂x − i∂f/∂y) dz + (1/2) (∂f/∂x + i∂f/∂y) dz̄.

According to this decomposition the limit f′(z_{0}) exists iff:

∂f/∂x (z_{0}) + ∂f/∂y (z_{0}) i = 0  (2)

(due to the nonexistence of the limit lim_{{z→z_{0} \atop z≠z_{0}}} (z̄ − z̄_{0})/(z − z_{0}) ).

Let’s consider: 1.) the set Γ = {γ : (−ϵ, ϵ) ⟶ C | ϵ > 0, γ (0) = 0, γ = γ_{1} + iγ_{2}, γ is not constant,

γ ∈ C^{1}, it exists and it is finite the limit:

\begin{aligned}L (γ) = & lim_{t→0} [(γ_{1}^{2}(t) − γ_{2}^{2}(t)) /(γ_{1}^{2}(t) + γ_{2}^{2}(t))+ \\ & − 2i (γ_{1}(t)γ_{2}(t)) /(γ_{1}^{2}(t) + γ_{2}^{2}(t)) ] \} \end{aligned}

2.) the equivalence relation “∼” on Γ:

γ ∼ δ ⟺ L (γ) = L (δ),

3.) the quotient set Γ / ∼ = Γ̃.

Definition 2.1. We call complex generalized derivative the following map:

D : C^{1} (Q ⊆ IR^{2}, IR^{2}) × Q × Γ̃ ⟶ C

\begin{matrix}D (f, z_{0}, γ̂) = & (1/2) [∂f/∂x (z_{0}) − (∂f/∂y (z_{0})) i]+ \\ + (1/2) [∂f/∂x (z_{0}) + (∂f/∂y (z_{0})) i] L (γ). \end{matrix}

Remark 2.2 One uses the condition (2) to be sure that the limit f′(z_{0}) exists for every sequence z_{n} ⟶ z_{0} (z_{n} ≠ z_{0}). If one accepts to have different limits of the function [f(z) − f(z_{0})] · (z − z_{0})^{−1}, for different sequences z_{n} ⟶ z_{0}, one can extend the definition of the complex derivative as in Definition 1.1.. Of course, the price of this extension is an extra index for the generalized complex derivative.

Let’s also remark that: L (γ) = lim_{t→0} [γ̄ (t) − γ̄ (0)] /[γ (t) − γ (0)], and if f is holomorph, then D (f, z_{0}, γ̂) = f′(z_{0}), ∀γ ∈ Γ.

Proposition 2.3. Let ϵ > 0 and γ : (−ϵ, ϵ) ⟶ C, γ (t) = a · t, a ∈ C^{∗}. Then γ ∈ Γ and L (γ) = γ̄′(0) /γ′(0).

Proof. We know that γ_{1}(t) = αt, γ_{2}(t) = βt and a = α + iβ. Hence:
L (γ) = lim_{t→0} [(α^{2}t^{2} − β^{2}t^{2}) /(α^{2}t^{2} + β^{2}t^{2}) − 2i(αtβt) /(α^{2}t^{2} + β^{2}t^{2})] =
= (α^{2} − β^{2})/(α^{2} + β^{2}) − 2i(αβ)/(α^{2} + β^{2}) =
= ā/a = γ̄′(0)/γ′(0).

Definition 2.4. We call rectilinear complex derivative, and we note it by D (f, z_{0}, a), a particular case of generalized complex derivative for γ (t) = a t, a ∈ C^{∗} (see Proposition 2.3.), therefore: D : C^{1}(Q ⊆ IR^{2}, IR^{2}) × Q × C^{∗} ⟶ C D (f, z_{0}, a) = D (f, z, γ̂), where γ (t) = a t, a ∈ C^{∗}.

Proposition 2.5. D (f, z_{0}, a) = ∂f/∂z (z_{0}) + (∂f/∂z̄(z_{0})) · ā/a, ∀a ∈ C^{∗}, ∀z_{0} ∈ Q, where we used Pompeiu’s notation (see [9]):

∂f/∂z (z_{0}) = (1/2) [∂f/∂x (z_{0}) − ∂f/∂y (z_{0})) i]
∂f/∂z̄(z_{0}) = (1/2) [∂f/∂x (z_{0}) + ∂f/∂y (z_{0})) i].

Proof.

D (f, z_{0}, a) = D (f, z_{0}, γ̂) =^{{def 1.1. \atop def 2.4.}} ∂f/∂z (z_{0}) + ∂f/∂z̄(z_{0})) L (γ) =

=^{prop 2.3.} ∂f/∂z (z_{0}) + (∂f/∂z̄(z_{0})) γ̄′(0) /γ′(0) =

= ∂f/∂z (z_{0}) + ∂f/∂z̄(z_{0})) ā/a.

Example 2.6. If f : C ⟶ C, f(z) = z̄, then D (f, z_{0}, a) = ā/a, ∀z_{0} ∈ C, ∀a ∈ C^{∗}. Example 2.7. If f : C ⟶ C, f(z) = |z|, then D (f, z_{0}, a) = 1/(2 |z_{0}|) = (z̄_{0} + z_{0} · ā/a).

3. Quaternionic derivative

Let’s consider: 1.) the set Γ = {γ : (−ϵ, ϵ) ⟶ H | ϵ > 0, γ (0) = 0, γ = γ_{1} + iγ_{2} + jγ_{3} + kγ_{4}, γ ∈ C^{1},

exists and are finite the following limits:

l_{11}(γ) = lim_{t→0} [γ_{1}^{2}(t) − γ_{2}^{2}(t) − γ_{3}^{2}(t) − γ_{4}^{2}(t)] / |γ (t)|^{2},
l_{22}(γ) = lim_{t→0} [−γ_{1}^{2}(t) + γ_{2}^{2}(t) − γ_{3}^{2}(t) − γ_{4}^{2}(t)] / |γ (t)|^{2},
l_{33}(γ) = lim_{t→0} [−γ_{1}^{2}(t) − γ_{2}^{2}(t) + γ_{3}^{2}(t) − γ_{4}^{2}(t)] / |γ (t)|^{2},
l_{ij}(γ) = lim_{t→0} [γ_{i}(t) · γ_{j}t] /|γ (t)|^{2}, ∀i, j ∈ {1, 2, 3, 4}, i ≠ j}

2.) the equivalence relation “∼” on Γ:

γ ∼ δ ⟺ l_{ij}(γ) = l_{ij}(δ) ∀i, j ∈ {1, 2, 3, 4},

3.) the quotient set Γ / ∼= Γ̃. In the followings Q is an open set in H (or in IR^{4}, since the topology is the same), w = x_{1} + ix_{2} + jx_{3} + kx_{4}, w_{0} ∈ Q, f = f_{1} + if_{2} + jf_{3} + kf_{4}.

194 S. LUGOJAN

Definition 2.8. We define the quaternionic derivative in the following manner: D : C^{1}(Q ⊆ IR^{4}, IR^{4}) × Q × Γ̃ ⟶ H

D (f, w_{0}, γ̂) =∂f/∂w (w_{o})+

+ (∂f/∂w̄ (w_{o})) [l_{11}(γ) − 2 il_{12}(γ) − 2 jl_{13}(γ) − 2 kl_{14}(γ)]+ + (∂f/∂w̃ (w_{o})) [l_{22}(γ) + 2 il_{21}(γ) − 2 jl_{24}(γ) − 2 kl_{23}(γ)]+ + (∂f/∂ŵ (w_{o})) [l_{33}(γ) − 2 il_{34}(γ) + 2 jl_{31}(γ) + 2 kl_{32}(γ)],

where we used the definition from [3], i.e.:

∂f/∂w (w_{0}) = (1/2) [∂f/∂x_{1} (w_{0}) − ∂f/∂x_{2} (w_{0})) i+

− (∂f/∂x_{3} (w_{0})) j + ∂f/∂x_{4} (w_{0}) k]

∂f/∂w̄ (w_{0})) = (1/2) [∂f/∂x_{1} (w_{0}) + (∂f/∂x_{4} (w_{0})) k]
∂f/∂w̃ (w_{0}) = (1/2) [−∂f/∂x_{2} (w_{0})) i + (∂f/∂x_{4} (w_{0})) k]
∂f/∂ŵ (w_{0}) = (1/2) [−∂f/∂x_{3} (w_{0})) j + (∂f/∂x_{4} (w_{0})) k].

Remark 2.9. One can write:

\begin{matrix}D (f, w_{0}, γ̂) = & ∂f/∂w (w_{0}) + (∂f/∂w̄ (w_{0})) l̄(γ)+ \\ + (∂f/∂w̃ (w_{0})) l̃(γ) + (∂f/∂ŵ (w_{0})) l̂(γ), \end{matrix}

using the notation:

\begin{aligned}l̄(γ) & = l_{11}(γ) − 2 il_{12}(γ) − 2 jl_{13}(γ) − 2 kl_{14}(γ) \\ l̃(γ) & = l_{22}(γ) + 2 il_{21}(γ) − 2 jl_{24}(γ) − 2 kl_{23}(γ) \\ l̂(γ) & = l_{33}(γ) − 2 il_{34}(γ) + 2 jl_{31}(γ) + 2 kl_{32}(γ). \end{aligned}

Let’s also remark that:

l̄(γ) = lim_{t→0} γ̄ (t) · γ^{−1}(t),
l̃(γ) = lim_{t→0} γ̃ (t) · γ^{−1}(t),
l̂(γ) = lim_{t→0} γ̂ (t) · γ^{−1}(t)

and since:

df = (∂f/∂w) dw + (∂f/∂w̄) dw̄ + (∂f/∂w̃) dw̃ + (∂f/∂ŵ) dŵ

(see [3]), it results that D is a hard type of derivative, i.e.: D (f, w_{0}, γ̂) = lim [f(w) − f(w_{0})] · (w − w_{0})^{−1}, where the limit is calculated when w tends to w_{o} on the curve t ⟶ w_{0} + γ (t). More of this, if f is H-derivable (see [3]), then D (f, w_{0}, γ̂) = f′(w_{0}), ∀γ ∈ Γ, where f′ is the quaternionic derivative defined in [3].

Proposition 2.10. If γ ∈ Γ, γ (t) = a · t, where a ∈ H^{∗}, a = a_{1} + ia_{2} + ja_{3} + ka_{4}, then: 1.) l_{11}(γ) = (a_{1}^{2} − a_{2}^{2} − a_{3}^{2} − a_{4}^{2}) /|a|^{2}

l_{22}(γ) = (−a_{1}^{2} + a_{2}^{2} − a_{3}^{2} − a_{4}^{2}) /|a|^{2} l_{33}(γ) = (−a_{1}^{2} − a_{2}^{2} + a_{3}^{2} − a_{4}^{2}) /|a|^{2} l_{ij}(γ) = a_{i} · a_{j} /|a|^{2}, ∀i, j ∈ {1, 2, 3, 4} i ≠ j,

2.) l̄(γ) = ā · a^{−1} = ā · ā · |a|^{−2},

l̃(γ) = ã · a^{−1} = ã · ā · |a|^{−2},

l̂(γ) = â · a^{−1} = â · ā · |a|^{−2}.

Proof. By straight calculation.

Definition 2.11. We call rectilinear quaternionic derivative the map:

D : C^{1}(Q ⊆ IR^{4}, IR^{4}) × Q × H^{∗} ⟶ H
D (f, w_{0}, a) = D (f, w_{0}, γ̂), where γ (t) = a · t a ∈ H^{∗}.

Proposition 2.12.

\begin{matrix}D (f, w_{0}, a) = & ∂f/∂w (w_{0}) + (∂f/∂w̄ (w_{0})) ā · a^{−1}+ \\ + (∂f/∂w̃ (w_{0})) ã · a^{−1} + (∂f/∂ŵ (w_{0})) â · a^{−1}. \end{matrix}

Proof. From Remark 2.9. and Proposition 1.2..

Example 2.13. 1.) If f : H ⟶ H, f(w) = w̄, then D (f, w_{0}, a) = ā · a^{−1}, ∀w_{0} ∈ H. 2.) If f : H ⟶ H, f(w) = w̃, then D (f, w_{0}, a) = ã · a^{−1}, ∀w_{0} ∈ H. 3.) If f : H ⟶ H, f(w) = ŵ, then D (f, w_{0}, a) = â · a^{−1}, ∀w_{0} ∈ H.

Example 2.14. If f : H ⟶ H, f(w) = w^{2}, then

\begin{matrix}D (f; x_{1} + ix_{2} + jx_{3} + kx_{4}; a) = & 2 x_{1} + 2 ix_{2} + 2 jx_{3} + (ix_{2} + jx_{3}) āa^{−1}+ \\ + (ix_{2} − kx_{4}) ãa^{−1} + (jx_{3} − kx_{4}) âa^{−1}. \end{matrix}

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