Phasors, trigonometric identities and periodic derivatives

For a real angle $u$, its unit phasor is $z(u):=\exp (iu)=e^{iu}$. On $\mathbb{R}$, define $u\sim v$ when $u-v=2\pi k$ for some $k\in \mathbb{Z}$. A phase is an equivalence class $[u]=\{v\in \mathbb{R}:u\sim v\}$ in the quotient set $\mathbb{R}\mathrm{/}\sim$; thus it is an angle considered modulo integer multiples of $2\pi$. An angular lift is a real representative of that class, or a locally real-valued representative when the phase varies. The functions $\sin$ and $\cos$ are the usual restrictions to real arguments of the complex trigonometric functions. For $w=a+ib$ with real $a,b$, write $\mathrm{Re}w=a$, $\mathrm{Im}w=b$, and $\bar{w}=a-ib$.

The following identities hold for real $u,v$: $z(u)=\cos u+i\sin u$, $\mathrm{\mid }z(u)\mathrm{\mid }=1$, $z(u-v)=z(u)\overline{z(v)}$, $\mathrm{Re}z(u-v)=\cos (u-v)$, and $\mathrm{Im}z(u-v)=\sin (u-v)$. With respect to an angular lift, $z^{\mathrm{\prime }}(u)=iz(u)$, ${\sin }^{\mathrm{\prime }}(u)=\cos u$, and ${\cos }^{\mathrm{\prime }}(u)=-\sin u$. These identities and derivatives are unchanged when any angle is replaced by a representative differing by an integer multiple of $2\pi$. Finite sums of differentiable complex-valued functions of a real variable may be differentiated termwise, and their real and imaginary parts are the respective sums of the real and imaginary parts.

Given: Real angles $u,v$, and, for the finite-sum assertion, finitely many complex-valued functions with differentiable real and imaginary components.

Goal: Establish the phase quotient, the displayed phasor identities, their independence of representatives and the finite-sum rules.

1.1 Set $x=0$ and $y=u$ in [F3]. Since $e^0=\exp (0)=1$, this gives $z(u)=\cos u+i\sin u$ and $\mathrm{\mid }z(u)\mathrm{\mid }=1$. By [F7], $z(v)\overline{z(v)}=1$. Also [F4] and [F1] give $z(v)z(-v)=1$, so $z(-v)=\overline{z(v)}$. Hence $z(u-v)=z(u)\overline{z(v)}$ by [F4], and the stated real and imaginary parts follow by applying the Cartesian formula at $u-v$. [F1, F3, F4, F7, algebra]

1.2 The relation $\sim$ is reflexive because $u-u=2\pi \cdot 0$. If $u-v=2\pi k$, then $v-u=2\pi (-k)$, proving symmetry. If $u-v=2\pi k$ and $v-w=2\pi \mathrm{\ell }$, then $u-w=2\pi (k+\mathrm{\ell })$, proving transitivity; the witnesses $0,-k,k+\mathrm{\ell }$ are integers. Hence [F8] supplies the quotient set of classes used to define phases, with real representatives as angular lifts. [F8, algebra]

2.1 By [F3] and [F4], $z(2\pi )=z(\pi {)}^2=1$. The addition law, $z(0)=1$, and induction in both integer directions imply $z(u+2\pi k)=z(u)$ for every integer $k$. Equality of the real and imaginary coordinates in step 1.1 makes sine and cosine $2\pi$-periodic as well. Consequently these functions have values independent of the representative of a phase in the quotient from step 1.2. [F1, F3, F4, step 1.1, step 1.2, algebra, induction]

3.1 Restricting [F5] to real arguments gives the stated sine and cosine derivatives. Differentiate the Cartesian expression in step 1.1 componentwise using [F6]; the result is $z^{\mathrm{\prime }}(u)=-\sin u+i\cos u=iz(u)$. The derivative formulas are periodic by step 2.1. A lift changed locally by a constant integer multiple of $2\pi$ therefore gives the same values and derivatives. [F5, F6, step 1.1, step 2.1, algebra]

4.1 For each summand write $f_j=a_j+i{b}_j$. Coordinate addition gives $\mathrm{Re}{\sum }_j{f}_j={\sum }_j{a}_j$ and $\mathrm{Im}{\sum }_j{f}_j={\sum }_j{b}_j$. Repeated application of the real sum rule in [F6] gives derivatives of these two finite sums; the complex difference quotient has these component quotients, so its limit is the sum of the component derivatives. This proves termwise differentiation and completes the justification. [F6, algebra, induction] ∎