It is proved that the function f (z) = x ^ 2 + 2xy-y ^ 2-I (x ^ 2-2xy-y ^ 2) is analytic everywhere in the complex plane and its derivative is obtained It is proved that the function f (z) = x ^ 2 + 2xy-y ^ 2-I (x ^ 2-2xy-y ^ 2) is analytic everywhere in the complex plane and its derivative is obtained,

It is proved that the function f (z) = x ^ 2 + 2xy-y ^ 2-I (x ^ 2-2xy-y ^ 2) is analytic everywhere in the complex plane and its derivative is obtained It is proved that the function f (z) = x ^ 2 + 2xy-y ^ 2-I (x ^ 2-2xy-y ^ 2) is analytic everywhere in the complex plane and its derivative is obtained,

The first method is to use the Cauchy Riemann equation. The second method is to directly form the function f (z) = (x + Yi) ^ 2-2xyi + 2xy-i [(x + Yi) ^ 2-2xyi-2xy] = (x + Yi) ^ 2-2xyi + 2xy-i (x + Yi) ^ 2-2xy + 2xyi = (x + Yi) ^ 2-I (x + Yi) ^ 2 = (1-I) Z ^ 2, so it is the analytic function f of complex plane everywhere