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User:Jheald/sandbox/GA/Cauchy-Riemann add

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To possibly add to Cauchy–Riemann equations, under #Generalisations:

Clifford analysis

[edit]

If the complex numbers are considered to be the even part of the Clifford algebra C2,0(R), which is spanned by the bases {1, e1, e2, i=e1e2} where e12=e22=1 and (e1e2)2 = -1, then the Cauchy–Riemann operator can be identified as e1 times the Dirac operator (the vector derivative using the Clifford product):

The Cauchy–Riemann equations can therefore be identified with a vector-field equation:

Taking even and odd parts of the Clifford product gives:

cf #Physical interpretation above.

In turn, this can be related to a (real) scalar potential function,

so that

Where does the minus sign in -v come from?

[edit]

Clifford algebra isn't commutative, so

Instead,

Therefore

i.e. (equating imaginary parts):

so it has to be