Viète's formula
The product formula of Vieta from 1593 is one of the first historically proven analytical representations for the district number. It is an infinite product with nested roots.
- 2.1 Analytical proof
- 2.2 Historical reasoning by Vieta
- 2.3 proof of the product-free representation
Representations of
Formula of Vieta
By the by
Recursively defined sequence of numbers applies:
Advertised with the first factors the infinite product is:
Relation to representation by Euler
The product formula of Vieta arises as a special case of the following result of Euler (see proof below) by inserting:
In particular, this results in the following alternative, direct method for the members of the sequence ( see above):
Free product representation
The following diagram is equivalent to the product formula of Vieta and has a simple geometric interpretation ( see, for example). With the result recursively defined
As well as building on the consequences and
Applies:
The first terms of the sequence are:
The followers are each just the side length and the sequence elements corresponding to half the circumference of the regular pentagon. And because of the associated numerical extinction in the display of the result of numerical calculation is not suitable.
Evidence
Analytical proof
Follows the one hand, by using the known limit
On the other hand, is obtained by iteratively applying the duplication formula for the sine:
Summarizing these two statements then leads to the representation of Euler:
So especially for:
Induction can now easily show that the cosine terms correspond with the terms of the sequence defined recursively:
For the equality follows immediately from the well-known special value of the cosine and (induction step ), use the halving formula for the cosine.
Historical reasoning by Vieta
The above analytical proof of Vieta's product formula based on the representation of, a result that Euler until over 100 years later and knew that was not yet Vieta available. His argument is geometric in nature and is a variation of Exhaustionsverfahren to calculate the circular area, which goes back to Archimedes. Starting from a square () used Vieta a sequence of regular vertices, which are inscribed in the unit circle and gradually approximate the area. The lengths and conditions required in the doubling receives Vieta by elementary geometric considerations (for example, using the Pythagorean theorem ).
Proof of the product-free representation
Through reciprocal of Education and multiplication by 2 follows from the product formula Vietaschen immediately following product formula for:
The claim for the product-free representation is obviously true, if for a sequence
Applies. This is easy to show by induction ( in this case go only the definitions of the consequences, and a, cf ).
A proof fully executed can be found for example in the proof archive, see Related links.