Giovanni Vacca (mathematician)

Italian mathematician- Giovanni Enrico Eugenio Vacca (18 November 1872 â€“ 6 January 1953) was an Italian mathematician, Sinologist and historian of science.
- Vacca studied mathematics and graduated from the University of Genoa in 1897 under the guidance of G.
- B.
- Negri.
- He was a politically active student and was banished for that from Genoa in 1897.
- He moved to Turin and became an assistant to Giuseppe Peano.
- In 1899 he studied, at Hanover, unpublished manuscripts of Gottfried Wilhelm Leibniz, which he published in 1903.
- Around 1898 Vacca became interested in Chinese language and culture after attending a Chinese exhibition in Turin.
- He took private lessons of Chinese and continued to study it at the University of Florence.
- Vacca then traveled to China in 1907â€“8 and defended a PhD in Chinese studies in 1910.
- In 1911, he became a lecturer in Chinese literature at the University of Rome.
- In 1922, he moved to Florence and taught Chinese literature and language at university until 1947.
- The interests of Vacca were almost equally split between mathematics, Sinology and history of science, with a corresponding number of papers being 38, 47 and 45.
- In 1910, Vacca developed a complex number iteration for pi: x 0 = i , x n + 1 = x n + | x n | 2 , lim n ? 8 x n = 2 p . {\displaystyle x_{0}=i,\quad x_{n+1}={\frac {x_{n}+|x_{n}|}{2}},\qquad \lim _{n\to \infty }x_{n}={\frac {2}{\pi }}.} The calculation efficiency of these formulas is significantly worse than of the modern Borwein's algorithm â€“ they converge by only about half a decimal point with each iteration. Vacca published his two major contributions to mathematics in 1910 and 1926, on series expansion (later named Vacca series) of the Euler constant.
- They are, respectively ? = ? k = 2 8 ( - 1 ) k ? log 2 ? k ? k = 1 2 - 1 3 + 2 ( 1 4 - 1 5 + 1 6 - 1 7 ) + 3 ( 1 8 - ? - 1 15 ) + ? ? ( 2 ) + ? = ? k = 2 8 ( 1 ? k ? 2 - 1 k ) = ? k = 2 8 k - ? k ? 2 k ? k ? 2 = 1 2 + 2 3 + 1 2 2 ( 1 5 + 2 6 + 3 7 + 4 8 ) + 1 3 2 ( 1 10 + ? + 6 15 ) + ? {\displaystyle {\begin{aligned}\gamma &=\sum _{k=2}^{\infty }(-1)^{k}{\frac {\left\lfloor \log _{2}k\right\rfloor }{k}}={\tfrac {1}{2}}-{\tfrac {1}{3}}+2\left({\tfrac {1}{4}}-{\tfrac {1}{5}}+{\tfrac {1}{6}}-{\tfrac {1}{7}}\right)+3\left({\tfrac {1}{8}}-\cdots -{\tfrac {1}{15}}\right)+\cdots \\[6pt]\zeta (2)+\gamma &=\sum _{k=2}^{\infty }\left({\frac {1}{\lfloor {\sqrt {k}}\rfloor ^{2}}}-{\frac {1}{k}}\right)=\sum _{k=2}^{\infty }{\frac {k-\lfloor {\sqrt {k}}\rfloor ^{2}}{k\lfloor {\sqrt {k}}\rfloor ^{2}}}\\[4pt]&={\tfrac {1}{2}}+{\tfrac {2}{3}}+{\tfrac {1}{2^{2}}}\left({\tfrac {1}{5}}+{\tfrac {2}{6}}+{\tfrac {3}{7}}+{\tfrac {4}{8}}\right)+{\tfrac {1}{3^{2}}}\left({\tfrac {1}{10}}+\cdots +{\tfrac {6}{15}}\right)+\cdots \end{aligned}}} Vacca noted in 1910 that: There is some hope that this series can be of some use in the proof of the irrationality of ? {\displaystyle \gamma } , a very difficult problem, proposed, but not resolved, in the Correspondence, recently published, between Hermite und Stieltjes.

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