Quaternion derivatives

[Topolfractal]

I am trying to work out some basic aspects in the theory of quaternions for some work in physics I'm doing. I have went through complex analysis and saw that the only way division ( and hence the derivative) could be defined was through a numerical definition of (i). My question is does there exist a notion of a quaternion derivative even without a numerical definition of ( k, quaternion part) ? I have some pieces of a possible answer, but am far away from meaningful mathematical consistency. I would greatly appreciate if someone would push me in the right direction.

 

[Mark44]

I am trying to work out some basic aspects in the theory of quaternions for some work in physics I'm doing. I have went through complex analysis and saw that the only way division ( and hence the derivative) could be defined was through a numerical definition of (i).

?
Division of complex numbers is done by multiplying by 1 in the form of the complex conjugate of the divisor over itself. This produces a real number in the denominator. What do you mean by "numerical definition of (i)"?

Also, what do you mean by the derivative here? Are you talking about a function f whose domain and range are the complex numbers?

My question is does there exist a notion of a quaternion derivative even without a numerical definition of ( k, quaternion part) ? I have some pieces of a possible answer, but am far away from meaningful mathematical consistency. I would greatly appreciate if someone would push me in the right direction.

I suppose you could have a function whose domain and range are quaternions. I have never heard about the derivative of such a function, but that doesn't mean that no one has done any work in this direction. To calculate a derivative you would need to use the difference quotient definition of the derivative, which entails doing division. This wikipedia article talks about the conjugate and the reciprocal of a quaternion - https://en.wikipedia.org/wiki/Quaternion

 

[Topolfractal]

That helps out a lot thank you. What I mean is that the complex conjugate and complex number become real under multiplication, because we know precisely what i=sqrt(-1) and how to multiply it. This is coming though from a person who heard about quaternions but not researched them thoroughly, so I could be completely wrong.

 

[Mark44]

That helps out a lot thank you. What I mean is that the complex conjugate and complex number become real under multiplication, because we know precisely what i=sqrt(-1) and how to multiply it.

I think I know what you mean, but that's not what you're saying. A complex number and its conjugate don't become real - their product is real, due to the way multiplication of complex numbers is defined (which includes the definition of i * i = -1).

This is coming though from a person who heard about quaternions but not researched them thoroughly, so I could be completely wrong.

 

[Topolfractal]

Thank for the clarifications and after skimming the article I now know where I went wrong and know now about Hamiltonian's insight. It's all making sense now.

I think I know what you mean, but that's not what you're saying. A complex number and its conjugate don't become real - their product is real, due to the way multiplication of complex numbers is defined (which includes the definition of i * i = -1).

Thank you that's where I was trying to go, but just couldn't phrase it right.

 

[blueberry]

The best and simplest way to compute a quaternionic derivative is as follows. We represent a quaternion argument p = x+y⋅i+z⋅j+u⋅k (i,j,k are basic quaternion units; "⋅" is the quaternion multiplication) and a quaternion-differentiable (holomorphic) function of that argument ψ(p) = ψ₁(x,y,z,u)+ψ₂(x,y,z,u)⋅i+ψ₃(x,y,z,u)⋅j+ψ₄(x,y,z,u)⋅k in the Cayley–Dickson doubling form: p = a+b⋅j, where a´= x+y⋅i ; b = z+u⋅i and ψ(p)=ψ(a,b)=Φ₁(a,b)+Φ₂(a,b)⋅j, where Φ₁(a,b)=ψ₁(a,b)+ψ₂(a,b)⋅i and Φ₂(a,b)=ψ₃(a,b)+ψ₄(a,b)⋅i. Each expression for ψ(p) is initially to be obtained from a complex function of the same kind by means of the direct replacement of a complex variable with a quaternion variable in the expression for the complex function. For example, ψ(p)=p^-1. Just as a complex- holomorphic function satisfies Cauchy-Riemann's equations in complex analysis, a quaternion- holomorphic function satisfies the following quaternionic generalization of Cauchy-Riemann's equations:

(1) ∂_aΦ₁ = ∂_b^×Φ₂^×, (2) ∂_aΦ₂ = - ∂_b^×Φ₁^×,
(3) ∂_aΦ₁ = ∂_bΦ₂, (4) ∂_a^×Φ₂ = - ∂_b^×Φ₁

after doing a = a^× = x,​

where the complex conjugation is denoted by × and the partial differentiation with respect to some variable s is denoted by ∂_s. For example, by ∂_b^×Φ₂^× is denoted the partial derivative of the complex conjugate of a function Φ₂ with respect to the complex conjugate of a complex variable b. Firstly, we compute the partial derivatives of functions Φ₁, Φ₂, Φ₁^×, Φ₂^× (with respect to variables a, b, a^×, b^×); secondly, we put a = a^× =x in the computed expressions of partial derivatives; and thirdly, we check whether equations (1) - (4) hold. One of the formulae to compute the first quaternionic derivative of the quaternion-holomorphic function is the following:

ψ(p)⁽¹⁾ = (∂_aΦ₁ + ∂_a^×Φ₁) + (∂_aΦ₂ + ∂_a^×Φ₂)⋅j .​

Higher derivatives of quaternion-holomorphic functions can be computed analogically and they are holomorphic like the first derivative.
For details and examples I refer to http://vixra.org/abs/1609.0006