Rasing a quaternion to a quaternionic power ?

[David Makin]

Hi all, this is a topic that I researched on the WWW quite a bit when I first played with quaternions and I never found a definitive answer to the question about doing:

 q^p = exp(p*log(q))  and/or  q^p = exp(log(q)*p)

Given Paolo's use of quaternions with respect to the triplex I'd love to hear people's opinions on the above.
I think I'm correct that the order doesn't matter if p is real or complex but if there is a j or k component to the power then the order of the multiplication does matter.

My own opinion is that both orders are equally valid but there are varied opinions - I even saw one person suggest that both calculations should be used and the result should be the average of the two ! And someone else suggested both should be used but the result chosen based on some geometric criteria of the original value of q and the results.

[fractalrebel]

Hi all, this is a topic that I researched on the WWW quite a bit when I first played with quaternions and I never found a definitive answer to the question about doing:

 q^p = exp(p*log(q))  and/or  q^p = exp(log(q)*p)

Given Paolo's use of quaternions with respect to the triplex I'd love to hear people's opinions on the above.
I think I'm correct that the order doesn't matter if p is real or complex but if there is a j or k component to the power then the order of the multiplication does matter.

My own opinion is that both orders are equally valid but there are varied opinions - I even saw one person suggest that both calculations should be used and the result should be the average of the two ! And someone else suggested both should be used but the result chosen based on some geometric criteria of the original value of q and the results.

Since quaternion multiplication does not commute I suspect you will get different results.

[fractalrebel]

Hi all, this is a topic that I researched on the WWW quite a bit when I first played with quaternions and I never found a definitive answer to the question about doing:

 q^p = exp(p*log(q))  and/or  q^p = exp(log(q)*p)

Given Paolo's use of quaternions with respect to the triplex I'd love to hear people's opinions on the above.
I think I'm correct that the order doesn't matter if p is real or complex but if there is a j or k component to the power then the order of the multiplication does matter.

My own opinion is that both orders are equally valid but there are varied opinions - I even saw one person suggest that both calculations should be used and the result should be the average of the two ! And someone else suggested both should be used but the result chosen based on some geometric criteria of the original value of q and the results.

Since quaternion multiplication does not commute I suspect you will get different results.

In my quaternion object library in UF5, I have both power functions. One is nameed Power and the other Power2. Dave, since you have access to the libraries, they are in reb.ulb.

[David Makin]

@Ron: I know, I think it was me that suggested both should be included in the library

@Paolo, Kerry and other mathematicians
I'm just wondering if one method is formally accepted as the "correct" one or if it's generally accepted that both are equally valid ?
(I mean from an academic standpoint)
It would also be interesting to know if there are any real physical phenomena that relate to the apparent choice of two results in this manner (e.g. orientation of photons or fundamental particles??).

[Paolo Bonzini]

@Paolo, Kerry and other mathematicians
I'm just wondering if one method is formally accepted as the "correct" one or if it's generally accepted that both are equally valid ?
(I mean from an academic standpoint)

I honestly have no idea. I'm pretty sure one is usually written as p^q and the other is explicitly written as exp & log, but I don't remember which. I think you are overestimating my mathematician-ness. ;-)