Euler's Identity and the Unifying Thread of Mathematics

<历史背景> The formula $e^{i\pi} + 1 = 0$, now universally known as Euler’s identity, is far more than a curiosity of complex analysis: it is the first explicit manifestation of the unifying thread that runs through nearly every branch of modern mathematics, a thread that stretches from the infinite series of 18th-century analysis to the topological invariants of 21st-century geometry, and from the arithmetic of prime numbers to the algebraic structures underlying quantum field theory. To understand its significance, we must first situate it in the historical and intellectual context of Leonhard Euler’s 1748 masterpiece *Introductio in Analysin Infinitorum* (Introduction to the Analysis of the Infinite), the work that formalized the field of complex analysis and laid the groundwork for much of modern mathematical physics.

Euler, born in Basel in 1707, was already one of the most prolific mathematicians in history by the time he published the *Introductio* at the age of 41. By 1735, he had lost the sight of his right eye due to overwork, and by 1766, he was completely blind, yet he continued to produce nearly half of his total 800+ published works after losing his sight entirely, relying on his prodigious memory and ability to perform complex calculations in his head. The *Introductio* was written during his first tenure at the St. Petersburg Academy, where he had been invited by Catherine I of Russia in 1727. At the time, the field of analysis was in a state of rapid but unrigorous development: Newton and Leibniz’s calculus had been formalized over the preceding half-century, but the foundations of infinite series were still poorly understood, and complex numbers were widely regarded as “fictitious” or “impossible” quantities, useful only as intermediate steps in solving cubic equations but with no standalone mathematical legitimacy.

Euler’s central goal in the *Introductio* was to unify the disparate threads of 17th-century mathematics: the calculus of Newton and Leibniz, the infinite series work of Newton, Leibniz, and the Bernoulli family, and the trigonometric and exponential functions that had previously been treated as entirely separate objects. He achieved this by introducing the exponential function $e^z$ as the unique function satisfying $f'(z) = f(z)$ with $f(0)=1$, and then deriving its power series expansion, a result that would later be recognized as the foundation of complex analysis. It was in this context that he first wrote down the relation between the exponential and trigonometric functions that we now call Euler’s formula, and from it derived the special case $e^{i\pi} + 1 = 0$, a result he presented almost as an afterthought, without recognizing the full depth of its unifying implications.

<原始推导与严格证明> Euler’s original 1748 derivation of the identity was characteristically heuristic, relying on formal manipulations of infinite series without the rigorous convergence theory that would not be developed for another half-century. He began with the power series expansions of the exponential, sine, and cosine functions, which he had derived earlier in the *Introductio*: $$ e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \frac{z^4}{4!} + \dots = \sum_{n=0}^\infty \frac{z^n}{n!} $$ $$ \cos z = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \frac{z^6}{6!} + \dots = \sum_{n=0}^\infty \frac{(-1)^n z^{2n}}{(2n)!} $$ $$ \sin z = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + \dots = \sum_{n=0}^\infty \frac{(-1)^n z^{2n+1}}{(2n+1)!} $$ Euler observed that if we substitute $z = i\theta$ (where $i = \sqrt{-1}$ is the imaginary unit, and $\theta$ is a real number) into the exponential series, the even-powered terms exactly match the cosine series, while the odd-powered terms are exactly $i$ times the sine series. Formally, he wrote: $$ e^{i\theta} = \left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \dots\right) + i\left(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \dots\right) = \cos\theta + i\sin\theta $$ This is Euler’s formula, the general case from which the identity follows immediately by setting $\theta = \pi$, since $\cos\pi = -1$ and $\sin\pi = 0$, giving $e^{i\pi} = -1$, or $e

$e^{i\pi} = -1$, or $e^{i\pi} + 1 = 0$, the identity that now bears his name. Euler himself made no special comment on this result in the *Introductio*, treating it as a straightforward corollary of his series expansions rather than a landmark discovery; it would take nearly a century for the mathematical community to recognize its profound unifying power. The heuristic nature of his 1748 derivation, which relied on formal manipulation of infinite series without a rigorous theory of convergence, was typical of the era: the ratio test for series convergence would not be formalized until the work of Cauchy in the 1820s, and the modern definition of complex differentiability would not appear until Riemann’s 1851 doctoral thesis. To bridge the gap between Euler’s original argument and modern rigorous analysis, we first formalize the proof of Euler’s formula and identity using contemporary tools, before tracing their historical reception and exploring their far-reaching generalizations.

1.1 Strict Proof of Euler’s Identity via Taylor Series Convergence

The foundation of Euler’s 1748 derivation lies in the power series expansions of the exponential, sine, and cosine functions, which he derived earlier in the *Introductio* by extending the single-variable Taylor series (first formalized by Brook Taylor in 1715) to complex arguments. For a complex-valued function $f(z)$ analytic at $z=0$, the Taylor series expansion is given by: $$f(z) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} z^n$$ where $f^{(n)}(0)$ denotes the $n$-th derivative of $f$ evaluated at 0. Applying this to the three core functions: 1. For $f(z) = e^z$, all derivatives satisfy $f^{(n)}(z) = e^z$, so $f^{(n)}(0) = 1$ for all $n$, yielding: $$e^z = \sum_{n=0}^\infty \frac{z^n}{n!} = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \dots$$ 2. For $f(z) = \cos z$, the derivatives cycle as $f^{(n)}(z) = \cos(z + n\pi/2)$, so $f^{(2n)}(0) = (-1)^n$ and $f^{(2n+1)}(0) = 0$, yielding: $$\cos z = \sum_{n=0}^\infty \frac{(-1)^n z^{2n}}{(2n)!} = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \dots$$ 3. For $f(z) = \sin z$, the derivatives cycle as $f^{(n)}(z) = \sin(z + n\pi/2)$, so $f^{(2n)}(0) = 0$ and $f^{(2n+1)}(0) = (-1)^n$, yielding: $$\sin z = \sum_{n=0}^\infty \frac{(-1)^n z^{2n+1}}{

$e^{i\pi} + 1 = 0$, the identity that now bears his name. Euler himself made no special comment on this result in the *Introductio*, treating it as a straightforward corollary of his series expansions rather than a landmark discovery; it would take nearly a century for the mathematical community to recognize its profound unifying power. The heuristic nature of his 1748 derivation, which relied on formal manipulation of infinite series without a rigorous theory of convergence, was typical of the era: the ratio test for series convergence would not be formalized until the work of Cauchy in the 1820s, and the modern definition of complex differentiability would not appear until Riemann’s 1851 doctoral thesis. To bridge the gap between Euler’s original argument and modern