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actual evaluation – Are these variations of advanced multiplication studied subjects? Answer

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actual evaluation – Are these variations of advanced multiplication studied subjects?

Complex multiplication may be very nicely understood geometrically and algebraically, however I marvel what concerning the following operators -angles assumed to breathe randians $[0,2pi)$:

  1. Complex multiplication(muladd): $$ x_1 cdot x_2 = |x_1||x_2|e^{(arg(x_1) + arg(x_2))i}$$
  2. Complex mulmul: $$ x_1 bigodot x_2 = |x_1||x_2|e^{arg(x_1) cdot arg(x_2)i} $$
  3. Complex addadd: $$ x_1 bigoplus x_2 = (|x_1|+|x_2|)e^{(arg(x_1) + arg(x_2))i} $$
  4. Complex addmul: $$ x_1 bigotimes x_2 = (|x_1|+|x_2|)e^{(arg(x_1) cdot arg(x_2))i} $$

Have these operators been studied? Are there any books or papers on their properties? Do they’ve names?

Note: I couldn’t discover a complex-analysis tag.

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