A First Course in Complex Analysis · Problem 2.5.14

Differentiation — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · cauchy-riemann-equations

Problem

Verify the Cauchy--Riemann equations for the following functions, and compute their derivatives: ((a)) f(z)=z3f(z) = z^3 ((b)) f(z)=1zf(z) = \dfrac{1}{z} ((c)) f(z)=ezf(z) = e^z ((d)) f(z)=z2+iz+3−2if(z) = z^2 + iz + 3 - 2i
Intuitive explanation

The Cauchy–Riemann equations are the gateway to differentiability in the complex plane: they tell you whether a function f(z)=u(x,y)+iv(x,y)f(z) = u(x,y) + iv(x,y) is actually holomorphic. Split each function into real and imaginary parts, check that ∂u∂x=∂v∂y\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} and ∂u∂y=−∂v∂x\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} hold everywhere (or everywhere except isolated points), then use f′(z)=∂u∂x+i∂v∂xf'(z) = \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x} to read off the derivative directly—no limit definition needed.

Intuitive explanation

The Cauchy–Riemann equations are the gateway to differentiability in the complex plane: they tell you whether a function f(z)=u(x,y)+iv(x,y)f(z) = u(x,y) + iv(x,y) is actually holomorphic. Split each function into real and imaginary parts, check that ∂u∂x=∂v∂y\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} and ∂u∂y=−∂v∂x\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} hold everywhere (or everywhere except isolated points), then use f′(z)=∂u∂x+i∂v∂xf'(z) = \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x} to read off the derivative directly—no limit definition needed.

Related theory

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