Complex numbers and Euler's formula

Intuition [Beginner]

A complex number is a point in the plane. The horizontal position is a regular number (the real part) and the vertical position is a regular number times a special symbol $i$, where $i^2=-1$. So $3+2i$ means the point three units right and two units up.

Euler's formula says that $e^{i\theta }=\cos \theta +i\sin \theta$. In plain language: if you walk angle $\theta$ around a circle of radius $1$ in the plane, the point you reach has coordinates $(\cos \theta ,\sin \theta )$. Raising $e$ to an imaginary power traces out the circle.

This concept exists because many patterns in mathematics connect exponential growth with rotation. Sound waves, electrical signals, and quantum-mechanical amplitudes all use this link. Without complex numbers, these connections remain hidden.

Visual [Beginner]

The picture shows the unit circle in the plane. A point sits on the circle at angle $\theta$ from the positive horizontal axis. Its horizontal distance from the centre is labelled $\cos \theta$ and its vertical distance is $\sin \theta$. An arrow from the origin to the point represents $e^{i\theta }$.

As $\theta$ increases from $0$ to $2\pi$, the point traces the full circle, returning to the starting position. Euler's formula turns multiplication by $e^{i\theta }$ into rotation by angle $\theta$.

Worked example [Beginner]

Compute $e^{i\pi }$.

Step 1. By Euler's formula, $e^{i\pi }=\cos \pi +i\sin \pi$.

Step 2. The cosine of $\pi$ is $-1$. The sine of $\pi$ is $0$.

Step 3. Therefore $e^{i\pi }=-1+i\cdot 0=-1$. Adding $1$ to both sides gives $e^{i\pi }+1=0$.

What this tells us: the five most important constants in mathematics — $e$, $i$, $\pi$, $1$, $0$ — are connected by a single equation.

Check your understanding [Beginner]

Formal definition [Intermediate+]

A complex number is an expression $z=a+bi$ where $a,b\in \mathbb{R}$ and $i^2=-1$. The set of complex numbers is $\mathbb{C}=\{a+bi:a,b\in \mathbb{R}\}$.

The real part is $\mathrm{Re}(z)=a$ and the imaginary part is $\mathrm{Im}(z)=b$. The complex conjugate is $\bar{z}=a-bi$. The modulus is $|z|=\sqrt{a^2+b^2}$. The argument $\arg (z)$ is the angle $\theta$ such that $z=|z|e^{i\theta }$, defined up to multiples of $2\pi$.

Definition (Complex exponential). For $z=x+iy\in \mathbb{C}$, the complex exponential is

$$e^z=e^x(\cos y+i\sin y).$$

Setting $x=0$ gives Euler's formula $e^{iy}=\cos y+i\sin y$.

Counterexamples to common slips [Intermediate+]

• The argument is not single-valued. For any non-zero $z$, there are infinitely many valid arguments: $\theta$ and $\theta +2\pi k$ give the same $z$ for every integer $k$. The principal value $\mathrm{Arg}(z)\in (-\pi ,\pi ]$ is a convention, not an intrinsic property.

• $e^z$ is not injective on $\mathbb{C}$. Unlike the real exponential $e^x$, the complex exponential satisfies $e^{z+2\pi i}=e^z$ for all $z$. This periodicity is the source of the complex logarithm's multivaluedness.

• $|e^{iy}|=1$, not $e^y$. The modulus of $e^{iy}=\cos y+i\sin y$ is $\sqrt{{\cos }^{2}y+{\sin }^{2}y}=1$, independent of $y$. The factor $e^x$ in $e^{x+iy}$ carries the modulus.

Key theorem with proof [Intermediate+]

Theorem (Euler's formula). For all $\theta \in \mathbb{R}$,

$$e^{i\theta }=\cos \theta +i\sin \theta .$$

Proof. Define three functions of $\theta$: $f(\theta )=e^{i\theta }$, $g(\theta )=\cos \theta$, $h(\theta )=\sin \theta$. Compute the derivative of $f$:

$$f^{\prime }(\theta )=\frac{d}{d\theta }{e}^{i\theta }=i{e}^{i\theta }=if(\theta ).$$

This follows from the power-series definition $e^{i\theta }={\sum }_{n=0}^{\infty }(i\theta {)}^n/n!$ and term-by-term differentiation, which is valid because the series converges absolutely.

Now consider $F(\theta )=f(\theta )\cdot e^{-i\theta }$. Differentiating:

$$F^{\prime }(\theta )=f^{\prime }(\theta )e^{-i\theta }+f(\theta )(-i)e^{-i\theta }=if(\theta )e^{-i\theta }-if(\theta )e^{-i\theta }=0.$$

Since $F^{\prime }=0$ on $\mathbb{R}$, the function $F$ is constant. At $\theta =0$: $F(0)=e^0\cdot e^0=1$. Therefore $f(\theta )=e^{i\theta }$ for all $\theta$.

To extract cosine and sine, separate into real and imaginary parts. Write $f(\theta )=u(\theta )+iv(\theta )$ where $u$ and $v$ are real-valued. Then $f^{\prime }=u^{\prime }+i{v}^{\prime }=i(u+iv)=-v+iu$, giving $u^{\prime }=-v$ and $v^{\prime }=u$. At $\theta =0$: $u(0)=1$, $v(0)=0$. The unique solution of this system with these initial conditions is $u(\theta )=\cos \theta$ and $v(\theta )=\sin \theta$. $\square$

Corollary (de Moivre's formula). For $n\in \mathbb{Z}$ and $\theta \in \mathbb{R}$:

$$(\cos \theta +i\sin \theta {)}^n=\cos (n\theta )+i\sin (n\theta ).$$

Proof. By Euler's formula, $\cos \theta +i\sin \theta =e^{i\theta }$. Then $(e^{i\theta }{)}^n=e^{in\theta }=\cos (n\theta )+i\sin (n\theta )$ by Euler's formula again. $\square$

Bridge. Euler's formula is the foundational reason that rotation in the plane and exponential growth are the same operation in the complex domain, and this is exactly the bridge between trigonometry and the exponential function. The central insight is that $i$ encodes a quarter-turn, so $e^{i\theta }$ encodes a turn by angle $\theta$. This result builds toward 02.04.04 where the Fourier expansion uses $e^{inx}$ to decompose periodic functions, and appears again in the Gamma function reflection formula where $\sin (\pi s)=(e^{i\pi s}-e^{-i\pi s})/(2i)$ connects Euler's formula to the Gamma function's analytic continuation. The pattern generalises to complex analysis, where $e^z$ is the first entire function and the exponential map appears in the universal cover of $\mathbb{C}\setminus \{0\}$.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (Euler's formula via power series). The identity $e^{i\theta }=\cos \theta +i\sin \theta$ follows from the Taylor series $e^{i\theta }={\sum }_{n=0}^{\infty }(i\theta {)}^n/n!$, $\cos \theta ={\sum }_{n=0}^{\infty }(-1{)}^n{\theta }^{2n}/(2n)!$, $\sin \theta ={\sum }_{n=0}^{\infty }(-1{)}^n{\theta }^{2n+1}/(2n+1)!$, and the relation $i^{2n}=(-1{)}^n$.

Theorem 2 (The $n$th roots of unity form a cyclic group). The set ${\mu }_n=\{e^{2\pi ik/n}:k=0,1,\dots ,n-1\}$ forms a cyclic group of order $n$ under multiplication, generated by $\omega =e^{2\pi i/n}$.

The roots of unity are equally spaced on the unit circle, dividing it into $n$ equal arcs. For $n$ prime, every non-identity element generates the group.

Theorem 3 (Complex logarithm). The complex logarithm $\log z=\ln |z|+i\arg z$ is multivalued: $\log z=\ln |z|+i(\mathrm{Arg}(z)+2\pi k)$ for $k\in \mathbb{Z}$. The principal branch $\mathrm{Log}(z)=\ln |z|+i\mathrm{Arg}(z)$ is holomorphic on $\mathbb{C}\setminus (-\infty ,0]$.

The branch cut on the negative real axis is the price of single-valuedness. Different branches are related by $2\pi i$, and the multivalued logarithm encodes the topology of $\mathbb{C}\setminus \{0\}$ as a space with fundamental group $\mathbb{Z}$.

Theorem 4 (Riemann surface of $\sqrt{z}$). The equation $w^2=z$ defines a two-sheeted covering of $\mathbb{C}\setminus \{0\}$. As $z$ traverses a loop around the origin, $w$ changes sign. The Riemann surface is topologically a plane (the map $w\mapsto w^2$ is a homeomorphism from the surface to $\mathbb{C}$).

Theorem 5 (Fundamental theorem of algebra via Liouville). Every non-constant polynomial $p\in \mathbb{C}[z]$ has a root in $\mathbb{C}$. The proof by Liouville's theorem: if $p$ has no zero, then $1/p$ is entire and bounded, hence constant — contradicting that $p$ is non-constant.

Theorem 6 (Exponential map as a covering). The map $\exp :\mathbb{C}\to \mathbb{C}\setminus \{0\}$ is a surjective holomorphic map with kernel $2\pi i\mathbb{Z}$. It is the universal covering of $\mathbb{C}\setminus \{0\}$, and the fundamental group ${\pi }_1(\mathbb{C}\setminus \{0\})\cong \mathbb{Z}$ is identified with the deck transformation group generated by $z\mapsto z+2\pi i$.

Synthesis. Euler's formula is the foundational reason that the complex exponential, trigonometric functions, and rotations in the plane are unified into a single object. The central insight is that $i$ is the algebraic encoding of a quarter-turn, and this is exactly the structure that identifies multiplication by $e^{i\theta }$ with rotation by angle $\theta$. Putting these together with de Moivre's formula and the roots of unity, the algebra of polynomial equations over $\mathbb{C}$ becomes transparent: every root of unity is a power of a single generator, and the cyclic group structure appears again in the topology of the punctured plane as the fundamental group ${\pi }_1(\mathbb{C}\setminus \{0\})$. The bridge is from algebra ($i^2=-1$) to geometry (rotation) to topology (covering space), and the pattern generalises to the Riemann surface of $\sqrt{z}$ where the same cyclic-group structure governs the branching. The exponential covering map builds toward 02.01.07 where fibrations and covering spaces are studied in full generality, and appears again in 02.11.08 where the Fourier transform decomposes functions into their frequency components via $e^{inx}$.

Full proof set [Master]

Proposition 1 (Euler's formula via power series). For all $\theta \in \mathbb{R}$, $e^{i\theta }=\cos \theta +i\sin \theta$.

Proof. Expand the exponential:

$$e^{i\theta }=\sum _{n=0}^{\infty }\frac{(i\theta {)}^n}{n!}=\sum _{n=0}^{\infty }\frac{i^n{\theta }^n}{n!}.$$

Split into even and odd powers. For $n=2k$: $i^{2k}=(i^2{)}^k=(-1{)}^k$. For $n=2k+1$: $i^{2k+1}=i\cdot (-1{)}^k$. Therefore:

$$e^{i\theta }=\sum _{k=0}^{\infty }\frac{(-1{)}^k{\theta }^{2k}}{(2k)!}+i\sum _{k=0}^{\infty }\frac{(-1{)}^k{\theta }^{2k+1}}{(2k+1)!}=\cos \theta +i\sin \theta .$$

Both series converge absolutely for all $\theta$ (ratio test with limit $0$), so the rearrangement is justified. $\square$

Proposition 2 (The Riemann surface of $w^2=z$ is simply connected). The surface $S=\{(z,w)\in {\mathbb{C}}^2:w^2=z\}$ is homeomorphic to $\mathbb{C}$.

Proof. Define $\varphi :\mathbb{C}\to S$ by $\varphi (w)=(w^2,w)$. This is well-defined since $w^2=z$ holds. Define $\psi :S\to \mathbb{C}$ by $\psi (z,w)=w$. Then $\psi (\varphi (w))=w$ and $\varphi (\psi (z,w))=(w^2,w)=(z,w)$, so $\varphi$ and $\psi$ are mutually inverse. Both are continuous (coordinate functions are polynomials). Hence $S\cong \mathbb{C}$ as topological spaces. Since $\mathbb{C}$ is simply connected, $S$ is simply connected. $\square$

Connections [Master]

• Fundamental theorems of calculus 02.04.04. Euler's formula is proved by differentiating $f(\theta )=e^{i\theta }$ and verifying that $f^{\prime }=if$. This derivative computation relies on the term-by-term differentiation of power series, which is justified by the uniform convergence guaranteed by the FTC framework. The exponential function $e^x$ is itself defined via the integral ${\int }_1^{e}dx/x$, connecting Euler's formula back to the fundamental theorem.

• Hilbert space and Fourier analysis 02.11.08. The Fourier basis functions $e^{inx}$ for $n\in \mathbb{Z}$ are orthogonal in $L^2([0,2\pi ])$, and every square-integrable periodic function decomposes as $f(x)={\sum }_{n=-\infty }^{\infty }{c}_n{e}^{inx}$. Euler's formula is the starting point: it provides the exponential functions whose orthogonality drives the entire Fourier theory.

• Fibration 02.01.07. The exponential map $\exp :\mathbb{C}\to \mathbb{C}\setminus \{0\}$ is a covering map with fibre $2\pi i\mathbb{Z}$, which is the simplest example of a fibre bundle with discrete fibre. The multivaluedness of the complex logarithm is the analytic manifestation of the non-simply-connected topology of $\mathbb{C}\setminus \{0\}$, and the universal cover provided by $\exp$ resolves the ambiguity.

• Dirichlet $L$-functions 21.03.02. The codomain ${\mathbb{C}}^{\times }$ of a Dirichlet character is the multiplicative group of non-zero complex numbers of the present unit, and the values of a Dirichlet character lie on the unit circle (since $|\chi (a)|=1$ for $\gcd (a,m)=1$), making each character a homomorphism into the group of $\phi (m)$-th roots of unity. The complex-analytic continuation of $L(s,\chi )$ to $\mathrm{Re}(s)\le 1$ uses the standard complex-analysis machinery — contour integration, residue theorem, Phragmén-Lindelöf principle — built on the foundations introduced here.

Historical & philosophical context [Master]

Euler 1748, in his Introductio in analysin infinitorum ^[Euler1748], derived the formula $e^{i\theta }=\cos \theta +i\sin \theta$ by equating the power series of the exponential with the alternating series of cosine and sine. The identity $e^{i\pi }+1=0$ appeared in this context as a special case, unifying the five fundamental constants. Euler treated complex numbers as formal objects; the geometric interpretation of $\mathbb{C}$ as a plane came later.

Gauss 1831, in a review in the Gottingische gelehrte Anzeigen ^[Gauss1831], gave the first authoritative justification of the complex plane as a geometric object, establishing that complex numbers correspond to points in ${\mathbb{R}}^2$ and that their algebraic operations have geometric meanings (addition is translation, multiplication by $r{e}^{i\theta }$ is scaling by $r$ and rotation by $\theta$). Riemann 1851, in his inaugural dissertation ^{[Riemann1851]}, introduced Riemann surfaces to handle multivalued functions like $\sqrt{z}$ and $\log z$, showing that the branch points and sheets of these surfaces encode the topological structure of the complex plane minus a point.

Bibliography [Master]

@book{Euler1748,
  author    = {Euler, Leonhard},
  title     = {Introductio in analysin infinitorum},
  publisher = {Marcum-Michaelem Bousquet},
  year      = {1748}
}

@article{Gauss1831,
  author  = {Gauss, Carl Friedrich},
  title   = {Anzeige der Theoria residuorum biquadraticorum, Commentatio secunda},
  journal = {G\"ottingische gelehrte Anzeigen},
  year    = {1831}
}

@phdthesis{Riemann1851,
  author = {Riemann, Bernhard},
  title  = {Grundlagen f\"ur eine allgemeine Theorie der Functionen einer ver\"anderlichen complexen Gr\"osse},
  school = {G\"ottingen},
  year   = {1851}
}

@book{Ahlfors1979,
  author    = {Ahlfors, Lars V.},
  title     = {Complex Analysis},
  publisher = {McGraw-Hill},
  edition   = {3},
  year      = {1979}
}

@book{Apostol1967,
  author    = {Apostol, Tom M.},
  title     = {Calculus, Vol. 1},
  publisher = {Wiley},
  year      = {1967}
}

@book{Needham1997,
  author    = {Needham, Tristan},
  title     = {Visual Complex Analysis},
  publisher = {Oxford University Press},
  year      = {1997}
}