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was when we learned that the branch line for the square root function is typically set along the negative reals.

bear with me

multiplying complex numbers involves rotating about the origin - so a square root consists of halving the angle they make with the positive reals. this leads to a conflict where, ... somewhat predictably ... there are two ways to halve an angle, so there are two square roots to any number. so you have to choose a convention, and a "branch line" where that convention wraps over - typically the negative real axis. "above" this axis, you halve angles clockwise, and below counter-clockwise. and on the branch line, the function is... undefined

so the conclusion here is you can't take the square root of negative 1 in complex analysis


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