The Complex Exponential and Euler's Formula

Set $\mathbb C=\mathbb R^2$, write $(x,y)=x+iy$, embed $x\in\mathbb R$ as $(x,0)$, and put $i=(0,1)$. Define $(x+iy)+(u+iv)=(x+u)+i(y+v),\qquad (x+iy)(u+iv)=(xu-yv)+i(xv+yu).$ Then $i^2=-1$. The conventions and prerequisite facts used below are recorded in The real numbers.

$\mathbb C$ with the preceding operations is a field. For $z=x+iy\ne0$, $z^{-1}=\frac{x-iy}{x^2+y^2}.$ The conventions and prerequisite facts used below are recorded in The complex numbers as $\mathbb R^2$, with their arithmetic, real embedding, and imaginary unit, The reals form a totally ordered field, Field.

Fix $z\in\mathbb C$. Apply The recursion theorem to the set $\mathbb C$, the initial value $1$, and the function $w\mapsto wz$. This defines the natural powers uniquely by

Let $\jmath\colon\mathbb N\to\mathbb Z$ be the embedding of The naturals embed in the integers. For an integer $r\ge0$, that lemma gives a unique $n\in\mathbb N$ with $r=\jmath(n)$; define $z^r:=z^n$. If $r<0$ and $z\ne0$, there is a unique $n\in\mathbb N$ with $n\ge1$ and $-r=\jmath(n)$; define

The inverse exists because $\mathbb C$ is a field by The complex numbers form a field, and every nonzero $x+iy$ has inverse $(x-iy)/(x^2+y^2)$. Thus nonnegative integer powers are defined for every $z$, while negative integer powers are defined exactly when $z\ne0$; in particular $0^0=1$ and no negative power of $0$ is defined. The integer and its unique natural representative are never conflated.

For $z,w\in\mathbb C$ and $n\in\mathbb N$, write $\iota_{\mathbb C}:\mathbb N\to\mathbb C$ for the canonical-natural map of The canonical natural $\iota(n) = n \cdot 1_F$ of a field. Then $(z+w)^n=\sum_{k\le n}\iota_{\mathbb C}\!\left(\binom nk\right) z^kw^{n-k}.$ The conventions and prerequisite facts used below are recorded in The complex numbers form a field, and every nonzero $x+iy$ has inverse $(x-iy)/(x^2+y^2)$, Integer powers in the complex field, The set $[A]^{k}$ of $k$-element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$, Pascal's rule $\binom{n+1}{k+1} = \binom{n}{k} + \binom{n}{k+1}$, and the hockey-stick identity $\sum_{i \le n}\binom{i}{k} = \binom{n+1}{k+1}$, The product $g_0 g_1 \cdots g_{n-1}$ of a finite list in a monoid, by recursion, with the empty product ($n = 0$) equal to the identity, Generalised associativity: in a monoid the product of a finite list does not depend on the bracketing, and in a commutative monoid it does not depend on the order of the factors either, The principle of mathematical induction.

For $z=x+iy$, set $\operatorname{Re}z=x$, $\operatorname{Im}z=y$, $\overline z=x-iy$, and $|z|=\sqrt{x^2+y^2}\ge0$. The conventions and prerequisite facts used below are recorded in The complex numbers as $\mathbb R^2$, with their arithmetic, real embedding, and imaginary unit, Square roots exist: a unique $\sqrt{a} \ge 0$ with $(\sqrt{a})^2 = a$; the positives are $\{x^2 : x \neq 0\}$.

For complex $z,w$, conjugation respects sums and products, $z\overline z=|z|^2$, $|zw|=|z||w|$, and $|z+w|\le|z|+|w|$. The conventions and prerequisite facts used below are recorded in Real and imaginary parts, complex conjugation, and modulus, The complex numbers form a field, and every nonzero $x+iy$ has inverse $(x-iy)/(x^2+y^2)$, The $p$-norms $\lVert x\rVert_p$ for rational $p \ge 1$, and $\lVert x\rVert_\infty$, Each $\lVert\cdot\rVert_p$ is a norm on $\mathbb{R}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_\infty$ of the published metric-spaces page.

For $z=x+iy$ and $w=u+iv$, put

Under the identification $\mathbb C=\mathbb R^2$, this is exactly the metric $d_2$ induced by the Euclidean norm of The $p$-norms $\lVert x\rVert_p$ for rational $p \ge 1$, and $\lVert x\rVert_\infty$. It is a metric by Each $\lVert\cdot\rVert_p$ is a norm on $\mathbb{R}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_\infty$ of the published metric-spaces page, so the metric axioms are established rather than assumed.

Convergence in $\mathbb C$, Cauchy sequences in $\mathbb C$, and continuity of maps between subsets of $\mathbb C$ mean the notions of Convergence of a sequence in a metric space: $x_k \to x$ iff $d(x_k, x) \to 0$ in $\mathbb{R}$, Cauchy sequence in a metric space, and Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$-$\delta$ form for $d_{\mathbb C}$ (and the restricted metric on a subset). These uses are therefore licensed by the metric-space definition Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric.

The metric space $\mathbb C$ is complete. A sequence $z_n=x_n+iy_n$ converges to $x+iy$ exactly when $x_n\to x$ and $y_n\to y$. The conventions and prerequisite facts used below are recorded in The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane, For $n \ge 1$ a sequence in $\mathbb{R}^n$ converges iff each coordinate sequence converges, is Cauchy iff each coordinate sequence is Cauchy, and $\mathbb{R}^n$ is complete in every norm.

A complex sequence is a function $a\colon\mathbb N\to\mathbb C$. Since the additive reduct $(\mathbb C,+,0)$ of the complex field is a commutative monoid, the finite-list construction of The product $g_0 g_1 \cdots g_{n-1}$ of a finite list in a monoid, by recursion, with the empty product ($n = 0$) equal to the identity, read additively, defines its complex partial sums by

The superscript is omitted when the summands already make the codomain clear.

The complex series $\sum a_n$ converges to $s\in\mathbb C$ when $S_N\to s$ in the metric of The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane; then $s$ is its sum. This is the same finite sum and convergence as the vector-series definition on $\mathbb R^2$ in Series of vectors in $\mathbb{R}^n$, absolute convergence, rearrangement, and the set of rearrangement sums. The series converges absolutely when the real series $\sum |a_n|$ of Series, partial sums, convergence and the sum, divergence, and the tail series converges. For a bijection $\sigma\colon\mathbb N\to\mathbb N$, its rearrangement along $\sigma$ is the complex series $\sum a_{\sigma(n)}$.

Let $(c_n)$ be a complex sequence and $a\in\mathbb C$. The complex power series centered at $a$ with coefficients $(c_n)$ is

with powers from Integer powers in the complex field and the preceding complex partial sums. To define its radius without ambiguity, form the specific real power series

centered at $0$. The radius of convergence of the complex power series is, by definition, the radius of this $Q$ in A real power series about a centre, its interval of convergence, and its radius in $[0,+\infty]$. Thus the coefficient sequence, real variable, and center of the comparison series are all explicit.

Every absolutely convergent complex series converges, and every rearrangement has the same sum. The conventions and prerequisite facts used below are recorded in Complex series, absolute convergence, complex power series, and radius of convergence, An absolutely convergent series in $\mathbb{R}^n$ converges, and every rearrangement converges to the same sum.

For $\sum_{n\ge0}c_n(z-a)^n$, set $L:=\limsup_{k\to\infty}|c_{k+1}|^{1/(k+1)}\in[0,+\infty],$ so no $0$th root occurs, and set $R:=\begin{cases}+\infty,&L=0,\\1/L,&0<L<+\infty,\\0,&L=+\infty.\end{cases}$ Then the series converges absolutely for $|z-a|<R$ and diverges for $|z-a|>R$; no assertion is made on $|z-a|=R$. At $z=a$ it converges to $c_0$, including when $R=0$. The conventions and prerequisite facts used below are recorded in Complex series, absolute convergence, complex power series, and radius of convergence, Every absolutely convergent complex series converges, and rearrangements preserve its sum, Limit superior and limit inferior of a real sequence as $\inf_n \sup_{k \ge n} x_k$ and $\sup_n \inf_{k \ge n} x_k$ in $\overline{\mathbb{R}}$, For finite $L$: $L = \limsup x_k$ iff for every $\varepsilon > 0$ one has $x_k < L + \varepsilon$ eventually and $x_k > L - \varepsilon$ frequently, Root test: $\limsup |a_k|^{1/k} < 1$ gives absolute convergence and hence convergence, $> 1$ gives divergence, and $= 1$ decides nothing.

If $\sum a_n=A$ and $\sum b_n=B$ converge absolutely in $\mathbb C$, and $c_n=\sum_{k\le n}a_kb_{n-k}$, then $\sum c_n$ converges absolutely and has sum $AB$. The conventions and prerequisite facts used below are recorded in Every absolutely convergent complex series converges, and rearrangements preserve its sum, The complex plane is complete, and convergence is equivalent to convergence of real and imaginary parts, The complex numbers as $\mathbb R^2$, with their arithmetic, real embedding, and imaginary unit, The product $g_0 g_1 \cdots g_{n-1}$ of a finite list in a monoid, by recursion, with the empty product ($n = 0$) equal to the identity, Generalised associativity: in a monoid the product of a finite list does not depend on the bracketing, and in a commutative monoid it does not depend on the order of the factors either, If $\sum a_k$ and $\sum b_k$ both converge absolutely then their Cauchy product converges absolutely, with sum $AB$, If $0 \le a_k \le b_k$ eventually, convergence of $\sum b_k$ gives convergence of $\sum a_k$, and divergence of $\sum a_k$ gives divergence of $\sum b_k$, Convergent series add and scale termwise, Conjugation laws, $z\overline z=|z|^2$, multiplicativity of modulus, and the triangle inequality.

Let $\iota_{\mathbb R}\colon\mathbb N\to\mathbb R$ be the canonical-natural map of The canonical natural $\iota(n) = n \cdot 1_F$ of a field, and let $j\colon\mathbb R\to\mathbb C$ be the real embedding $j(x)=x+i0$ from The complex numbers as $\mathbb R^2$, with their arithmetic, real embedding, and imaginary unit. For every $n\in\mathbb N$, the factorial $n!$ is nonzero by The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$, so $\iota_{\mathbb R}(n!)>0$ by Canonical naturals are positive and strictly increasing and consequently $j(\iota_{\mathbb R}(n!))\ne0$ in the complex field The complex numbers form a field, and every nonzero $x+iy$ has inverse $(x-iy)/(x^2+y^2)$.

For $z\in\mathbb C$, define

whenever this complex series converges. Inside a complex expression we abbreviate the embedded denominator $j(\iota_{\mathbb R}(n!))$ by $n!$, so the same definition may be written $\exp z=\sum z^n/n!$ without identifying a natural number with a complex number. Powers and series are those of Integer powers in the complex field and Complex series, absolute convergence, complex power series, and radius of convergence. The convergence for every $z\in\mathbb C$ is discharged by The complex exponential series converges absolutely for every complex argument ↗.

For every $z\in\mathbb C$, the series $\sum z^n/n!$ converges absolutely. The conventions and prerequisite facts used below are recorded in The complex exponential by its power series, The exponential series converges absolutely for every real argument, Every absolutely convergent complex series converges, and rearrangements preserve its sum.

For all $z,w\in\mathbb C$, $\exp(z+w)=\exp z\exp w$. For real $x$, the complex value $\exp(x+0i)$ equals the published real exponential $e^x$. The conventions and prerequisite facts used below are recorded in The complex exponential by its power series, The complex exponential series converges absolutely for every complex argument, The Cauchy product of two absolutely convergent complex series converges absolutely to the product of their sums, The real exponential function and the number $e$ by a power series, The binomial theorem over the complex field, $\binom{n}{k}\,k!\,(n-k)! = n!$ for $k \le n$; hence $\binom{n}{k}\,k! = n^{\underline{k}}$, the quotient $n!/(k!(n-k)!)$ is a natural number, and $\binom{n}{k} = \binom{n}{n-k}$.

For every real $\theta$, $\exp(i\theta)=\cos\theta+i\sin\theta$.

For real $x,y$, $\exp(x+iy)=e^x(\cos y+i\sin y)$ and $|\exp(x+iy)|=e^x$. In particular $e^{i\pi}+1=0$. The conventions and prerequisite facts used below are recorded in $\exp(z+w)=\exp z\,\exp w$, and the complex exponential extends the real exponential, Euler's formula: $\exp(i\theta)=\cos\theta+i\sin\theta$ for every real $\theta$, Conjugation laws, $z\overline z=|z|^2$, multiplicativity of modulus, and the triangle inequality, Pythagorean and parity identities for all six trigonometric functions on their natural domains, The exponential is positive and satisfies $\exp(-x)=1/\exp(x)$, Quarter-turn values and shifts by pi/2 and pi, Pi as twice the smallest positive zero of cosine.

$\ker(\exp)=2\pi i\mathbb Z,$ and $\exp z=\exp w$ exactly when $z-w\in2\pi i\mathbb Z$. The conventions and prerequisite facts used below are recorded in $\exp(x+iy)=e^x(\cos y+i\sin y)$, $|\exp(x+iy)|=e^x$, and $e^{i\pi}+1=0$, The exponential function is strictly increasing, The zero sets of sine and cosine and the least positive common period 2 pi, $t\mapsto(\cos t,\sin t)$ is a bijection from $[0,2\pi)$ onto the real unit circle.

Every $z\ne0$ has a unique representation $z=r(\cos\theta+i\sin\theta)$ with $r=|z|>0$ and $-\pi<\theta\le\pi$. The conventions and prerequisite facts used below are recorded in Real and imaginary parts, complex conjugation, and modulus, $t\mapsto(\cos t,\sin t)$ is a bijection from $[0,2\pi)$ onto the real unit circle, The zero sets of sine and cosine and the least positive common period 2 pi.

The complex exponential maps $\mathbb C$ onto $\mathbb C\setminus\{0\}$. The conventions and prerequisite facts used below are recorded in Every nonzero complex number has a unique polar form $r(\cos\theta+i\sin\theta)$ with $r>0$ and $-\pi<\theta\le\pi$, $\exp(x+iy)=e^x(\cos y+i\sin y)$, $|\exp(x+iy)|=e^x$, and $e^{i\pi}+1=0$, The natural logarithm as the inverse of the exponential function, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm.

Let $j:\mathbb Z\to\mathbb Q$ be the integer embedding of The integers embed in the rationals, let $\iota_{\mathbb R}:\mathbb Q\to\mathbb R$ be the ordered-field embedding of The unique embedding of ℚ into an ordered field, and put $\kappa_{\mathbb R}:=\iota_{\mathbb R}\circ j$. For every integer $m$ and real $\theta$, $(\cos\theta+i\sin\theta)^m=\cos(\kappa_{\mathbb R}(m)\theta)+i\sin(\kappa_{\mathbb R}(m)\theta).$ The conventions and prerequisite facts used below are recorded in Euler's formula: $\exp(i\theta)=\cos\theta+i\sin\theta$ for every real $\theta$, $\exp(z+w)=\exp z\,\exp w$, and the complex exponential extends the real exponential, The complex numbers form a field, and every nonzero $x+iy$ has inverse $(x-iy)/(x^2+y^2)$, Integer powers in the complex field.

Write $\iota_{\mathbb R}:\mathbb N\to\mathbb R$ for the canonical-natural map of The canonical natural $\iota(n) = n \cdot 1_F$ of a field. If $n\in\mathbb N$ with $n\ge1$ and $z=re^{i\theta}\ne0$, its $n$ distinct roots are $r^{1/n}\exp\!\left(i\frac{\theta+2\pi\iota_{\mathbb R}(k)}{\iota_{\mathbb R}(n)}\right),\qquad k\in\mathbb N,\quad 0\le k<n.$ For $z=0$, the only $n$th root is $0$. Thus the $n$th roots of unity are precisely $\exp\!\left(i\frac{2\pi\iota_{\mathbb R}(k)}{\iota_{\mathbb R}(n)}\right)$ for $k\in\mathbb N$ with $0\le k<n$. The conventions and prerequisite facts used below are recorded in Every nonzero complex number has a unique polar form $r(\cos\theta+i\sin\theta)$ with $r>0$ and $-\pi<\theta\le\pi$, Complex de Moivre formula for every integer exponent, $\ker(\exp)=2\pi i\mathbb Z$, and $\exp z=\exp w$ exactly when $z-w\in2\pi i\mathbb Z$, Existence and uniqueness of $n$-th roots: a unique $a^{1/n} \ge 0$ with $(a^{1/n})^n = a$, The canonical natural $\iota(n) = n \cdot 1_F$ of a field, Integer powers in the complex field.