A First Course in Complex Analysis · Problem 1.5.16

Complex Numbers — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · properties-of-exponential-notation

Problem

Prove Proposition 1.2.7: For any

\phi, \phi_1, \phi_2 \in \mathbb{R}

, ((a))

e^{i\phi_1}e^{i\phi_2} = e^{i(\phi_1+\phi_2)}

((b))

e^{i0} = 1

((c))

\dfrac{1}{e^{i\phi}} = e^{-i\phi}

((d))

e^{i(\phi+2\pi)} = e^{i\phi}

((e))

\left|e^{i\phi}\right| = 1

((f))

\dfrac{d}{d\phi}e^{i\phi} = i\,e^{i\phi}