Vieta's product representation of the circle number Pi

The product formula of Vieta from 1593 is one of the first historically proven analytical representations for the circle number . It is an infinite product with nested roots . ${\displaystyle \pi }$

Representations of ${\displaystyle \pi }$

Formula from Vieta

With the through

${\displaystyle {\begin{aligned}a_{1}&:={\frac {1}{2}}{\sqrt {2}}\\a_{n}&:={\frac {1}{2}}{\sqrt {2+2a_{n-1}}}\qquad n\geq 2\end{aligned}}}$

recursively defined sequence of numbers applies: ${\displaystyle a_{n}}$

${\displaystyle \lim _{n\to \infty }\prod _{i=1}^{n}a_{i}=a_{1}\cdot a_{2}\cdot a_{3}\cdots ={\frac {2}{\pi }}.}$

The infinite product is written out with the first factors:

${\displaystyle {\frac {2}{\pi }}=\left({\frac {1}{2}}{\sqrt {2}}\right)\cdot \left({\frac {1}{2}}{\sqrt {2+{\sqrt {2}}}}\right)\cdot \left({\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}\right)\cdots }$

Relationship to Euler's representation

The product formula of Vieta results as a special case from the following result from Euler (see proof below) by inserting : ${\displaystyle x={\tfrac {\pi }{2}}}$

${\displaystyle {\begin{aligned}{\frac {\sin(x)}{x}}&=\lim _{n\to \infty }\prod _{i=1}^{n}\cos \left({\frac {x}{2^{i}}}\right)=\cos \left({\frac {x}{2}}\right)\cdot \cos \left({\frac {x}{4}}\right)\cdot \cos \left({\frac {x}{8}}\right)\cdots \\{\frac {2}{\pi }}&=\lim _{n\to \infty }\prod _{i=1}^{n}\cos \left({\frac {\pi }{2^{i+1}}}\right)=\cos \left({\frac {\pi }{4}}\right)\cdot \cos \left({\frac {\pi }{8}}\right)\cdot \cos \left({\frac {\pi }{16}}\right)\cdots \end{aligned}}}$

In particular, this results in the following alternative, direct representation for the terms of the number sequence ( see above ): ${\displaystyle a_{n}}$

${\displaystyle a_{n}=\cos \left({\frac {\pi }{2^{n+1}}}\right)\qquad \qquad \mathrm {f{\ddot {u}}r} \;n\geq 1}$

Product-free representation

The following illustration is equivalent to the Vieta product formula and has a simple geometric interpretation (see for example). With the recursively defined sequence ${\displaystyle r_{n}}$

${\displaystyle {\begin{aligned}r_{0}&:=0\\r_{n}&:={\sqrt {2+r_{n-1}}}\qquad \qquad \mathrm {f{\ddot {u}}r} \;n\geq 1\end{aligned}}}$

as well as the consequences and based on this ${\displaystyle s_{n}}$ ${\displaystyle u_{n}}$

${\displaystyle s_{n}={\sqrt {2-r_{n-1}}}}$

${\displaystyle u_{n}=2^{n}\;s_{n}=2^{n}\;{\sqrt {2-r_{n-1}}}}$

applies:

${\displaystyle \lim _{n\to \infty }u_{n}=\lim _{n\to \infty }2^{n}{\sqrt {2-\underbrace {\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots +{\sqrt {2}}}}}}}} _{(n-1){\text{-fache Schachtelung}}}}}=\pi }$

The first members of the sequence are: ${\displaystyle u_{n}}$

${\displaystyle {\begin{aligned}u_{1}&=2\cdot {\sqrt {2}}\\u_{2}&=4\cdot {\sqrt {2-{\sqrt {2}}}}\\u_{3}&=8\cdot {\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}\end{aligned}}}$

${\displaystyle \vdots \qquad \qquad \vdots \qquad \qquad \vdots }$

The following elements are each just the length of the side and the following elements are half the circumference of the regular corner. Because of and the associated numerical cancellation in , the representation of by the sequence is not suitable for numerical calculation. ${\displaystyle s_{n}}$ ${\displaystyle u_{n}}$ ${\displaystyle 2^{n+1}}$ ${\displaystyle \lim _{n\to \infty }r_{n}=2}$ ${\displaystyle s_{n}}$ ${\displaystyle \pi }$ ${\displaystyle u_{n}}$

proofs

Analytical evidence

The proof outlined below is based on addition theorems from trigonometry and an elementary limit value consideration. A detailed proof can be found on Wikibooks . Out

${\displaystyle {\begin{aligned}2^{n}\cdot \sin \left({\frac {x}{2^{n}}}\right)&=x\cdot \left({\frac {\sin \left({\frac {x}{2^{n}}}\right)}{\frac {x}{2^{n}}}}\right)\end{aligned}}}$

follows on the one hand by using the known limit value ${\displaystyle \lim _{x\to 0}{\tfrac {\sin x}{x}}=1}$

${\displaystyle {\begin{aligned}\lim _{n\to \infty }2^{n}\cdot \sin \left({\frac {x}{2^{n}}}\right)&=x.\end{aligned}}}$

On the other hand, by iteratively applying the doubling formula for the sine :

${\displaystyle \sin x=2\cdot \sin \left({\frac {x}{2}}\right)\cos \left({\frac {x}{2}}\right)=\ldots =2^{n}\cdot \sin \left({\frac {x}{2^{n}}}\right)\cdot \prod _{i=1}^{n}\cos \left({\frac {x}{2^{i}}}\right)}$

Combining these two statements then leads to Euler's presentation :

${\displaystyle {\begin{aligned}{\frac {\sin x}{x}}&=\lim _{n\to \infty }\prod _{i=1}^{n}\cos \left({\frac {x}{2^{i}}}\right)\end{aligned}}}$

So especially for : ${\displaystyle x={\tfrac {\pi }{2}}}$

${\displaystyle {\begin{aligned}{\frac {2}{\pi }}&=\lim _{n\to \infty }\prod _{i=1}^{n}\cos \left({\frac {\pi }{2^{i+1}}}\right)\end{aligned}}}$

It can now easily be shown inductively that the cosine terms agree with the terms of the recursively defined sequence : ${\displaystyle a_{n}}$

For equality follows directly from the known special value of the cosine and for (induction step) the halving formula for the cosine is used . ${\displaystyle n=1}$ ${\displaystyle \cos({\tfrac {\pi }{4}})={\tfrac {1}{2}}{\sqrt {2}}}$ ${\displaystyle n>1}$

Historical reasoning according to Vieta

The above analytical proof for Vieta's product formula is based on the representation for , a result that Euler only knew more than 100 years later and which Vieta was not yet available. His reasoning is of a geometric nature and is a variation of the exhaustion method for calculating the area of ​​a circle, which goes back to Archimedes . Starting from a square ( ), Vieta uses a series of regular corners that are inscribed in the unit circle and successively approximate the area. Vieta gets the lengths and proportions required for doubling through elementary geometric considerations (for example using the Pythagorean theorem ). ${\displaystyle {\tfrac {\sin x}{x}}}$ ${\displaystyle n=2}$ ${\displaystyle 2^{n}}$

Proof of the product-free presentation

By calculating the reciprocal value and multiplying by 2, the Vietaschen product formula immediately results in the following product formula for : ${\displaystyle \pi }$

${\displaystyle \pi =\lim _{n\to \infty }2\cdot \prod _{i=1}^{n}{\frac {1}{a_{i}}}}$

The claim for the product-free representation is obviously true if for the number sequence ${\displaystyle u_{n}}$

${\displaystyle u_{n}=2\cdot \prod _{i=1}^{n}{\frac {1}{a_{i}}}}$

applies. This can easily by induction show (here go only the definitions of the consequences , , and one, see.). ${\displaystyle a_{n}}$ ${\displaystyle r_{n}}$ ${\displaystyle s_{n}}$ ${\displaystyle u_{n}}$

A fully executed evidence can be found, for example, in the evidence archive, see web links .

literature

• P. Beckmann: A History of Pi , St. Martin's Press, New York, New York, 1971, ISBN 978-0312381851
• L. Berggren, J. Borwein, P. Borwein: L Pi: A source book , Second Edition, Springer Verlag, New York, 2000, ISBN 978-0387949246
• Aaron Levin: A New Class of Infinite Products Generalizing Viète's Product Formula for ${\displaystyle \pi }$ , The Ramanujan Journal, Volume 10, Number 3, December 2005, pp. 305-324 (20), doi : 10.1007 / s11139-005-4852-z
• TJ Osler: The united Vieta's and Wallis's products for ${\displaystyle \pi }$ , American Mathematical Monthly, 106 (1999), pp. 774-776.
• TJ Osler and M. Wilhelm: Variations on Vieta's and Wallis's products for pi , Mathematics and Computer Education, 35 (2001), pp. 225-232.
• Heinrich Quillmann: Exercises using the Pythagorean Theorem for calculating ${\displaystyle \pi }$ . (Exercises with the Pythagorean theorem to determine the circle number.) (German), PM Prax. Math. Sch. 45, No. 6: 285 (2003).
• Franciscus Vieta : Variorum de Rebus Mathematicis Reponsorum Liber VII , (1593) in: Francisci Vietae Opera Mathematica, (reprinted) Georg Olms Verlag, Hildesheim, New York, 1970, pp. 398-400 and 436-446. ( Online version of the complete works Francisci Vietae Opera Mathematica available on the website of the ETH Library Zurich )

Web links

Wikibooks: Evidence of Vieta's Product Formula and Related Statements  - Learning and Teaching Materials

Individual evidence

1. ↑ see original work Franciscus Vieta , Variorum de Rebus Mathematics Reponsorum Liber VII, (1593) in: Francisci Vietae Opera Mathematica, (reprinted) Georg Olms Verlag, Hildesheim, New York, 1970, pp. 398-400 and 436-446. ( Online version of the complete works Francisci Vietae Opera Mathematica available on the website of the ETH Library Zurich )
2. ↑ see for example P. Beckmann, A History of Pi, St. Martin's Press, New York, New York, 1971, ISBN 978-0312381851
3. ↑ L. Berggren, J. Borwein, P. Borwein, Pi: A source book, Second Edition, Springer Verlag, New York, 2000, ISBN 978-0387949246
4. ↑ ^{a b} Heinrich Quillmann: Exercises using the Pythagorean Theorem for calculating . (Exercises with the Pythagorean theorem to determine the circle number.) (German), PM Prax. Math. Sch. 45, No. 6, 285 (2003) ${\displaystyle \pi }$