So, is there a quaternion complex calculus?

In summary, the conversation discusses the possibility of an analogue of "complex calculus" for complex numbers using quaternions. This would involve defining quaternions using "commutation relations" and exploring the equivalents of "Cauchy-Riemann" equation and "cauchy integral formula" on this space. The conversation also mentions the potential usefulness of this concept in studying relativity and provides a link to a website with more information on quaternions.
  • #1
lokofer
106
0
"Quaternion complex calculus?"..

Hello my question is if there exist an analogue of "complex calculus" for complex numbers but using "quaternions"..in fact if we define the Quaternion.. [tex] a+ib+jc+kd [/tex] with the "commutation relations" [tex] [x_i ,x_j ]=\epsilon _{ijk} x_k [/tex] then i would like to know if there exist the equivalent to..

a) "Cauchy-Riemann" equation...( Harmonic functions ?)

b) "cauchy integral formula" (for Surface, Volume or line integrals on this space)

Of course the can be very 2fruitful" specially when studying Relativity since you could consider the Quaternion:

[tex] Q= cdt-idx-jdy-kdz [/tex] (x,y,z) space coordinates and t the time.
 
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  • #2
There is a nice website that maintains info and links about http://www.greenspun.com/bboard/q-and-a.tcl?topic=quaternions" .
 
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  • #3
Much of what's interesting is simply vector analysis in disguise. It would be good to look at that.
 
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Related to So, is there a quaternion complex calculus?

1. What is quaternion complex calculus?

Quaternion complex calculus is a branch of mathematics that deals with the study of functions and their derivatives using quaternions, which are a type of hypercomplex numbers that extend the complex numbers. It combines concepts from both complex analysis and quaternion algebra to solve problems in three-dimensional space.

2. How is it different from traditional complex calculus?

The main difference between quaternion complex calculus and traditional complex calculus is that quaternion calculus uses 4-dimensional quaternions instead of 2-dimensional complex numbers. This allows for the study of functions and their derivatives in three dimensions, rather than just two.

3. What are some applications of quaternion complex calculus?

Quaternion complex calculus has many applications in engineering and physics, particularly in the fields of robotics, computer graphics, and quantum mechanics. It can be used to model and analyze the motion of objects in three-dimensional space and to solve problems involving rotations and orientations.

4. Is it difficult to learn and understand?

Like any branch of mathematics, quaternion complex calculus can be challenging to learn and understand. It requires a solid understanding of both complex numbers and quaternions, as well as a strong foundation in calculus. However, with dedication and practice, it can be mastered.

5. Are there any resources available to learn quaternion complex calculus?

Yes, there are many resources available to learn quaternion complex calculus, including textbooks, online courses, and tutorials. Some recommended resources include "Quaternion Calculus and Fast Animation" by Ken Shoemake and "Applied Quaternionic Analysis" by Leiba Rodman and Ian M. Sigal.

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