Euler’s Formula

Summary: §138 of Chapter 8. Combining De Moivre’s formula $(\cos z\pm i\sin z{)}^n=\cos nz\pm i\sin nz$ with the Chapter 7 limit $(1+z/j{)}^j=e^z$, Euler arrives at

$\cos v=\frac{e^{iv}+e^{-iv}}{2},\sin v=\frac{e^{iv}-e^{-iv}}{2i},e^{iv}=\cos v+i\sin v.$

Sines and cosines are real linear combinations of complex exponentials, and the complex exponential is determined by its real and imaginary parts — sine and cosine. The promised connection of §126 between trigonometric and exponential transcendentals is realized.

Sources: chapter8 (§138)

Last updated: 2026-04-27

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The derivation

Take the §133 De Moivre representations

$\cos nz=\frac{(\cos z+i\sin z{)}^n+(\cos z-i\sin z{)}^n}{2},\sin nz=\frac{(\cos z+i\sin z{)}^n-(\cos z-i\sin z{)}^n}{2i}.$

Apply the §134 substitution: let $z$ be infinitely small, $n=j$ infinitely large, $jz=v$ finite. Then

$\cos z=1,\sin z=z=v/j,$

so

$\cos z\pm i\sin z=1\pm iv/j.$

Substituting into the §133 expressions:

$\cos v=\frac{(1+iv/j{)}^j+(1-iv/j{)}^j}{2},\sin v=\frac{(1+iv/j{)}^j-(1-iv/j{)}^j}{2i}.$

Now invoke the Chapter 7 identity $(1+w/j{)}^j=e^w$ for $j$ infinite. Setting $w=iv$ in one factor and $w=-iv$ in the other:

$(1+iv/j{)}^j=e^{iv},(1-iv/j{)}^j=e^{-iv}.$

Therefore

$\cos v=\frac{e^{iv}+e^{-iv}}{2},\sin v=\frac{e^{iv}-e^{-iv}}{2i}.$

(source: chapter8, §138).

Solving for the exponential

Compute $\cos v+i\sin v$:

$\cos v+i\sin v=\frac{e^{iv}+e^{-iv}}{2}+i\cdot \frac{e^{iv}-e^{-iv}}{2i}=\frac{e^{iv}+e^{-iv}}{2}+\frac{e^{iv}-e^{-iv}}{2}=e^{iv}.$

Likewise $\cos v-i\sin v=e^{-iv}$. So Euler’s formula:

$\boxed{e^{iv}=\cos v+i\sin v,e^{-iv}=\cos v-i\sin v.}$

(source: chapter8, §138). Euler comments: “From these equations we understand how complex exponentials can be expressed by real sines and cosines.”

Why the formula is forced

The identity is not posited — it is the only way three earlier results can coexist:

1. The §133 De Moivre identity, which says $(\cos z+i\sin z{)}^n$ rotates by $nz$.
2. The §122 / §125 identity $(1+w/j{)}^j=e^w$ for $j$ infinite.
3. The §134 infinitesimal substitution, which maps $(\cos z+i\sin z{)}^n$ to $(1+iv/j{)}^j$ when $z=v/j$ is infinitesimal and $n=j$ infinite.

Stack these and there is exactly one possibility: the rotation $\cos v+i\sin v$ equals $e^{iv}$. Euler does not present the formula as a deep insight; he writes it down at the end of a five-line computation.

This is also consistent with the Chapter 7 §125 hint: $\log (\cos z+i\sin z)$ should equal $iz$, since $e^{iz}=\cos z+i\sin z$.

Equivalent forms

• $e^{iv}+e^{-iv}=2\cos v$
• $e^{iv}-e^{-iv}=2i\sin v$
• $\sin v=(e^{iv}-e^{-iv})/(2i)$
• $\cos v=(e^{iv}+e^{-iv})/2$
• $\tan v=-i(e^{iv}-e^{-iv})/(e^{iv}+e^{-iv})$
• $\cot v=i(e^{iv}+e^{-iv})/(e^{iv}-e^{-iv})$

These will be used freely throughout later chapters of the Introductio. In particular §139 inverts the formula to express the arc itself as a logarithm.

What about $e^{i\pi }=-1$?

Setting $v=\pi$ in Euler’s formula:

$e^{i\pi }=\cos \pi +i\sin \pi =-1+0i=-1,$

i.e., $e^{i\pi }+1=0$. The “Euler identity” — celebrated as the most beautiful equation in mathematics — is an immediate corollary. Euler does not single it out in §138 — the formula is just one of many consequences he reads off — and the slogan “most beautiful equation” is a much later marketing flourish. But the substance is here.

How the formula is used in the rest of Chapter 8

• §139–§140 inverts Euler’s formula to express the arc as $z=(1/2i)\log \frac{\cos z+i\sin z}{\cos z-i\sin z}$.
• §139 shows that every logarithm of a complex number is a real number plus an arc, foreshadowing the full theory of complex logarithms in later sections.
• §142 uses the inverse to compute $\pi$ from rapidly convergent rational arctangent series.

In later chapters of the Introductio, Euler’s formula recurs constantly — every factorization of a polynomial with complex roots, every Fourier-style decomposition, every bridge between exponential growth and oscillation passes through this identity.

Related pages

• de-moivre-formula
• exponential-series
• sine-and-cosine-series
• infinitesimal-and-infinite-numbers
• eulers-number
• arctangent-series
• sine-and-cosine
• chapter-8-on-transcendental-quantities-which-arise-from-the-circle
• chapter-7-on-exponentials-and-logarithms-expressed-through-series