Proof of Periodicity of the Complex Exponential and Its Kernel

Routine arithmetic in $\mathbb{R}$, $\mathbb{Z}$ and $\mathbb{C}$ is used without comment (Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity). Let $\cos$ and $\sin$ be the cosine and sine functions. Everything rests on Euler's formula, which reduces questions about $\exp$ on the imaginary axis to the values and the periodicity of $\cos$ and $\sin$.

Special values. By Euler's formula Euler's Formula: the Complex Exponential on the Imaginary Axis, Its Modulus and Its Conjugate (i),

$$\exp(it)=\cos t+i\sin t\qquad\text{for every real }t.$$

For $t=\tfrac{\pi}{2}$ and $t=\pi$, the values in Quarter-Turn Identities and Periodicity of Sine and Cosine (ii) give

$$\exp\bigl(\tfrac{\pi}{2}i\bigr)=0+1\cdot i=i,\qquad \exp(\pi i)=-1+0\cdot i=-1 ;$$

for $t=2\pi$, Quarter-Turn Identities and Periodicity of Sine and Cosine (vi) with $m=1$ gives $\cos2\pi=1$ and $\sin2\pi=0$, so $\exp(2\pi i)=1$. This is (i).

The period. For every $z\in\mathbb{C}$, the functional equation The Complex Exponential: the Functional Equation, Conjugates, Real Arguments and the Modulus (ii) and the last value in (i) give

$$\exp(z+2\pi i)=\exp z\cdot\exp(2\pi i)=\exp z .$$

This is (ii).

The kernel: integer multiples of $2\pi i$ are sent to $1$. If $z=2\pi ki$ with $k\in\mathbb{Z}$, we write $z=i\,(2\pi k)$ with the real number $2\pi k$; then Euler's Formula: the Complex Exponential on the Imaginary Axis, Its Modulus and Its Conjugate (i) with $t=2\pi k$ and Quarter-Turn Identities and Periodicity of Sine and Cosine (vi) with $m=k$ give

$$\exp z=\cos(2\pi k)+i\sin(2\pi k)=1+0\cdot i=1 .$$

The kernel: nothing else is sent to $1$. Conversely, let $\exp z=1$, and write $z=x+iy$ with real numbers $x$ and $y$ (The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers (vii)). We show first that $x=0$ and then that $y$ is an integer multiple of $2\pi$.

To see that $x=0$, take moduli: by The Complex Exponential: the Functional Equation, Conjugates, Real Arguments and the Modulus (vi) and Basic Properties of the Exponential Function (i),

$$\exp_{\mathbb{R}}(x)=|\exp z|=|1|=1=\exp_{\mathbb{R}}(0).$$

Since $\exp_{\mathbb{R}}(u)<\exp_{\mathbb{R}}(v)$ whenever $u<v$ (Basic Properties of the Exponential Function (ix)), neither $x<0$ nor $0<x$ is possible; hence $x=0$.

Now $z=iy$, so Euler's formula Euler's Formula: the Complex Exponential on the Imaginary Axis, Its Modulus and Its Conjugate (i) gives $1=\exp(iy)=\cos y+i\sin y$. Since also $1=1+0\cdot i$, the uniqueness of the real coefficients $a$ and $b$ in the form $a+bi$ of The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers (vii) yields

$$\cos y=1,\qquad \sin y=0 .$$

We reduce $y$ modulo $2\pi$. As $2\pi>0$ by Quarter-Turn Identities and Periodicity of Sine and Cosine (i), the number $\frac{y}{2\pi}$ is defined, and by The Integers in the Real Numbers: Greatest Elements, the Integer Part, and Archimedean Consequences (iv) there is an integer $k$ with $k\le\frac{y}{2\pi}<k+1$. Put $s=y-2\pi k$, so that $0\le s<2\pi$. By Quarter-Turn Identities and Periodicity of Sine and Cosine (vi), applied at the real number $y$ with $m=-k$,

$$\cos s=\cos y=1,\qquad \sin s=\sin y=0 .$$

It remains to show that $s=0$, and we do so by excluding every other position of $s$ in $[0,2\pi)$. If $0<s<\pi$, then $\sin s>0$ by Quarter-Turn Identities and Periodicity of Sine and Cosine (vii). If $s=\pi$, then $\cos s=-1$ by Quarter-Turn Identities and Periodicity of Sine and Cosine (ii). If $\pi<s<2\pi$, then $0<s-\pi<\pi$, and Quarter-Turn Identities and Periodicity of Sine and Cosine (iv), applied at the real number $s-\pi$, together with Quarter-Turn Identities and Periodicity of Sine and Cosine (vii), gives

$$\sin s=\sin\bigl((s-\pi)+\pi\bigr)=-\sin(s-\pi)<0 .$$

Each of these cases contradicts $\cos s=1$ and $\sin s=0$. Hence $s=0$, so $y=2\pi k$ and $z=2\pi ki$. Together with the previous paragraph, this proves (iii).