Keywords

• complex analysis
• number theory
• Vieta formula
• Fabius function
• elliptic curve

Abstract We give two extensions of the classical Vieta (or Viète) formula $\displaystyle \dfrac{2}{\pi} = \dfrac{\sqrt{2}}{2} \dfrac{\sqrt{2 + \sqrt{2}}}{2} \dfrac{\sqrt{2 + \sqrt{2 + \sqrt{2}}}}{2} \cdots$ The first extension leads to the classical Fabius function, an infinitely differentiable function that is nowhere analytic. The second extension discusses the corresponding formula for the elliptic curves with complex multiplication $y^2 = x^3 -Dx,\quad y^2 = x^3 -D$.

We give two extensions of the classical Vieta (or Viète) formula $\displaystyle \dfrac{2}{\pi} = \dfrac{\sqrt{2}}{2} \dfrac{\sqrt{2 + \sqrt{2}}}{2} \dfrac{\sqrt{2 + \sqrt{2 + \sqrt{2}}}}{2} \cdots$ The first extension leads to the classical Fabius function, an infinitely differentiable function that is nowhere analytic. The second extension discusses the corresponding formula for the elliptic curves with complex multiplication $y^2 = x^3 -Dx,\quad y^2 = x^3 -D$.