A First Course in Complex Analysis · Problem 2.5.16

Differentiation — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · cauchy-riemann-equations

Problem

Find all points where the following functions satisfy the Cauchy--Riemann equations, and determine where they are holomorphic: ((a))

f(x+iy) = x^3 + 3xy^2 - 3x + i(y^3 + 3x^2y - 3y)

((b))

f(x+iy) = x^2 - y^2 - x + i(2xy - y)

((c))

f(x+iy) = e^x(\cos y - i \sin y)