A recovery of Brouncker's proof for the quadrature continued fraction.

Título inglés A recovery of Brouncker's proof for the quadrature continued fraction.
Título español Recuperación de la demostración de Brouncker de la fracción continua para la cuadratura.
Autor/es Khrushchev, Sergey
Organización Dep. Math. Atilim Univ., Incek (Ankara), Turquía
Revista 0214-1493
Publicación 2006, 50(1): 3-42, 23 Ref.
Tipo de documento articulo
Idioma Inglés
Resumen inglés 350 years ago in Spring of 1655 Sir William Brouncker on a request by John Wallis obtained a beautiful continued fraction for 4/π. Brouncker never published his proof. Many sources on the history of Mathematics claim that this proof was lost forever. In this paper we recover the original proof from Wallis' remarks presented in his Arithmetica Infinitorum. We show that Brouncker's and Wallis' formulas can be extended to MacLaurin's sinusoidal spirals via related Euler's products. We derive Ramanujan's formula from Euler's formula and, by using it, then show that numerators of convergents of Brouncker's continued fractions coincide up to a rotation with Wilson's orthogonal polynomials corresponding to the parameters a=0, b=1/2, c=d=1/4.
Clasificación UNESCO 120500 ; 120223
Palabras clave español Fracciones continuas ; Cuadratura ; Polinomios ortogonales ; Funciones especiales ; Historia de las matemáticas
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