Worked solution: Euler's proof via the sine product formula (1734–1735)

Frequently asked questions

Was Euler wrong to publish this before it was fully justified?

No — he checked the numerical value $\pi^2/6$ and later gave other derivations of the same product; the mathematical community accepted the result long before it was made fully rigorous.

What exactly does Weierstrass's theorem add?

It shows precisely which entire functions equal a convergent product over their zeros times $e^{g(x)}$ for some entire $g$ , and gives a growth condition (finite genus) under which $g$ must be constant — which holds for $\sin x$ .

Are there fully rigorous proofs in Euler's original style today?

Yes: the logarithmic derivative of the sine product matches the cotangent partial-fraction expansion, which complex analysis proves rigorously; and the double-integral proof of Beukers, Calabi and Kolk reaches the same identity with no infinite products at all.

Source

Leonhard Euler, 1735

Euler: The Master of Us All

Step 6 of 6: An honest gap, closed 140 years later

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In plain words

Euler had, in effect, cashed a check without first confirming the bank account existed: he assumed an "infinite polynomial" behaves like a finite one, without any proof that this is legitimate. The check did clear — but only because Karl Weierstrass built the missing bank, 140 years later, showing exactly which functions can be rebuilt from their zeros this way.

\sin x = x\prod_{n=1}^{\infty}\left(1-\frac{x^2}{n^2\pi^2}\right)

Detailed analysis

Euler's product for $\sin x$ is correct, and he later confirmed it by other, independent routes. But the leap in Step 3 — that a function is essentially determined by its zeros the way a polynomial is — only became a real theorem with Karl Weierstrass's factorisation theorem for entire functions (1876), which gives an explicit criterion for when no extra exponential factor is needed. This proof is presented largely as Euler gave it, gap and all, because that is how the identity was actually discovered.

Terms in this step

Weierstrass factorization theorem: A 1876 theorem of Karl Weierstrass showing exactly which entire functions equal a convergent product over their zeros times $e^{g(x)}$ for some entire function $g$ , and when $g$ must be constant.
Entire function: A function of a complex variable that is differentiable at every point of the complex plane, such as $\sin z$ or $e^z$ .

Knowledge used in this step

Functions of a complex variable