So, is there a quaternion complex calculus?

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SUMMARY

The discussion centers on the existence of a quaternion analogue to complex calculus, specifically exploring the Cauchy-Riemann equations and the Cauchy integral formula within the context of quaternions. The user inquires about harmonic functions and integral formulas applicable to quaternionic analysis, particularly in relation to applications in relativity. The conversation highlights the potential benefits of quaternion calculus in understanding physical phenomena, suggesting that it may provide deeper insights into vector analysis.

PREREQUISITES
  • Understanding of quaternion algebra, specifically the representation a + ib + jc + kd.
  • Familiarity with complex calculus concepts, including the Cauchy-Riemann equations.
  • Knowledge of vector analysis and its applications in physics.
  • Basic understanding of relativity and its mathematical frameworks.
NEXT STEPS
  • Research quaternion calculus and its applications in physics.
  • Study the Cauchy-Riemann equations in the context of quaternions.
  • Explore the Cauchy integral formula for quaternions and its implications.
  • Investigate the relationship between quaternion analysis and vector calculus.
USEFUL FOR

Mathematicians, physicists, and engineers interested in advanced calculus, particularly those exploring the applications of quaternions in relativity and vector analysis.

lokofer
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"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.. a+ib+jc+kd with the "commutation relations" [x_i ,x_j ]=\epsilon _{ijk} x_k 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:

Q= cdt-idx-jdy-kdz (x,y,z) space coordinates and t the time.
 
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There is a nice website that maintains info and links about http://www.greenspun.com/bboard/q-and-a.tcl?topic=quaternions" .
 
Last edited by a moderator:
Much of what's interesting is simply vector analysis in disguise. It would be good to look at that.
 
Last edited:

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