zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Inequalities for the perimeter of an ellipse. (English) Zbl 0985.26009
The authors describe a method to study whether an algebraic approximation to the perimeter of an ellipse is from above or below. By the representation of the perimeter in terms of hypergeometric functions the problem boils down to establishing the sign of the error $$ E(x)=F(1/2, -1/2; 1; x)-A(x) , $$ where $A(x)$ is an algebraic function (depending on the approximation chosen) of the parameter $x \in (0,1)$ related to the eccentricity of the ellipse. This problem can be tackled analyzing the sign of a series whose entries are all $>0$ starting from a sufficiently large index. Thus, the question is reduced to the sign of a polynomial given by the sum of a finite number of terms of the series. In the situation described its coefficients are integers, and we can apply a Sturm sequence argument with the aid of a computer algebra system performing integer arithmetics. In this way, the authors show that several classical formulas approximate the elliptical perimeter from below, proving in particular a conjecture by Vuorinen on a Muir’s formula.

26D07Inequalities involving other types of real functions
33C05Classical hypergeometric functions, ${}_2F_1$
33C75Elliptic integrals as hypergeometric functions
41A30Approximation by other special function classes
Full Text: DOI
[1] Almkvist, G.; Berndt, B.: Gauss, landen, Ramanujan, the arithmetic--geometric mean, ellipses,$ {\pi}$, and the ladies diary. Amer. math. Monthly 95, 585-608 (1988) · Zbl 0665.26007
[2] Berndt, B.: Ramanujan’s notebooks, part III. (1985) · Zbl 0555.10001
[3] Barnard, R. W.; Pearce, K.; Richards, K.: An inequality involving the generalized hypergeometric function and the arc length of an ellipse. SIAM J. Math. anal. 32, 403-419 (2000) · Zbl 0983.33006
[4] Devlin, K.: The logical structure of computer-aided mathematical reasoning. Amer. math. Monthly 104, 632-646 (1997) · Zbl 0891.00001
[5] Henrici, P.: Applied and computational complex analysis. (1974) · Zbl 0313.30001
[6] Vuorinen, M.: Hypergeometric functions in geometric function theory. Proceedings of special functions and differential equations (1998) · Zbl 0948.30024
[7] Young, D.; Gregory, R.: A survey of numerical mathematics. (1973) · Zbl 0262.65002