Tuesday, March 9, 2010

About Cauchy-Riemann Equation

Recently playing with complex numbers I discovered that transformation caused by matrix $$ \left( \begin{array}{cc} a_0 & -a_1 \\ a_1 & a_0 \end{array} \right) $$ is equivalent to multiplication over complex number $z=a_0+a_1i$. The Caushy-Riemann equation follows from this matrix.

Initially I assumed similar transformation for quaternion. However appropriate matrix for quaternion looks too restrictive. I assumed less restrictive condition for derivative of quaternion function $$ \frac {\partial y^0} {\partial x^0} = \frac{\partial y^1}{\partial x^1} = \frac{\partial y^2}{\partial x^2} = \frac{\partial y^3}{\partial x^3} $$ \[ \frac{\partial y^i}{\partial x^j} =- \frac{\partial y^j}{\partial x^i} \quad i\ne j \] I see that such functions like $y=ax$, $y=xa$, $y=x^2$ satisfy to this equations. The same time conjugation does not satisfy it. A lot of questions should be answered to understand if this is set of function that generalize set of complex functions in case of quaternions. You can find details in my paper eprint arXiv:0909.0855 Quaternion Rhapsody, 2010

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