Those are the Cauchy-Riemann equations, which determine when a function is analytic, or equivalently holomorphic.
One theorem in this field states that under certain hypotheses, if a function g is strictly smaller than a function f on the boundary of a region, then f and f plus g have the same number of zeroes in that region. The fundamental theorem of algebra can be proven as a corollary of a theorem from this field by arguing that the reciprocal of a polynomial with no zeroes would have to be bounded, and thus (*) constant. The main objects of study in this field satisfy the relations u sub x equals v sub y and u sub y equals negative v sub x. Those are the Cauchy-Riemann equations, which determine when a function is analytic, or equivalently holomorphic. For 10 points, name this branch of mathematics that extends the results of real analysis to functions over C.
One theorem in this field states that under certain hypotheses, if a function g is strictly smaller than a function f on the boundary of a region, then
complex analysis [prompt on just “analysis”]