arithmetic–geometric mean identities
E157393
Arithmetic–geometric mean identities are a collection of formulas and relationships that express various mathematical constants and special functions in terms of the arithmetic–geometric mean of two numbers.
All labels observed (5)
| Label | Occurrences |
|---|---|
| arithmetic–geometric mean | 3 |
| Gauss–Legendre algorithm | 2 |
| Gaussian arithmetic–geometric mean | 1 |
| arithmetic–geometric mean algorithm | 1 |
| arithmetic–geometric mean identities canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1382603 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: arithmetic–geometric mean identities Context triple: [Gauss’s constant, appearsIn, arithmetic–geometric mean identities]
-
A.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
B.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
C.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
D.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
E.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: arithmetic–geometric mean identities Target entity description: Arithmetic–geometric mean identities are a collection of formulas and relationships that express various mathematical constants and special functions in terms of the arithmetic–geometric mean of two numbers.
-
A.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
B.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
C.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
D.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
E.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical identity collection
ⓘ
result in analysis ⓘ special function identity ⓘ |
| field |
mathematical analysis
ⓘ
number theory ⓘ special functions ⓘ |
| hasKeyFormula |
K(1/\sqrt{2}) = \frac{\Gamma(1/4)^{2}}{4\sqrt{\pi}}
ⓘ
K(k) = \frac{\pi}{2\,\operatorname{AGM}(1,\sqrt{1-k^{2}})} ⓘ \frac{1}{\operatorname{AGM}(1,\sqrt{1-k^{2}})} = \frac{2}{\pi}K(k) ⓘ \int_{0}^{\infty}\frac{dx}{\sqrt{(x^{2}+a^{2})(x^{2}+b^{2})}} = \frac{\pi}{2\,\operatorname{AGM}(a,b)} ⓘ \int_{0}^{\pi/2}\frac{d\theta}{\sqrt{a^{2}\cos^{2}\theta + b^{2}\sin^{2}\theta}} = \frac{\pi}{2\,\operatorname{AGM}(a,b)} ⓘ \operatorname{AGM}(1,1/\sqrt{2}) = \frac{\Gamma(1/4)^{2}}{4\sqrt{\pi^{3}}} ⓘ \operatorname{AGM}(1,\sqrt{1-k^{2}}) = \frac{\pi}{4}\,\frac{1}{K(k)} ⓘ \operatorname{AGM}(1,\sqrt{1-k^{2}})\,\operatorname{AGM}(1,\sqrt{1-k'^{2}}) = \frac{\pi}{2} with k'^{2}=1-k^{2} ⓘ \operatorname{AGM}(a,b) = \frac{\pi}{4}\,\frac{a}{K(k)} with k^{2}=1-\left(\frac{b}{a}\right)^{2},\ a\ge b>0 ⓘ \pi = 2\,\operatorname{AGM}(1,1/\sqrt{2})^{2}\,\sum_{n=0}^{\infty}2^{n}(a_{n}^{2}-b_{n}^{2}) (Gauss–Legendre type) ⓘ \pi = 2\,\operatorname{AGM}(1,\sqrt{1-k^{2}})\,K(k) ⓘ |
| hasProperty |
connect arithmetic means, geometric means, and elliptic integrals
ⓘ
express many special values of elliptic integrals in closed form ⓘ provide rapidly convergent algorithms for computing elliptic integrals ⓘ provide rapidly convergent algorithms for computing pi ⓘ |
| historicalOrigin |
19th century theory of elliptic integrals
ⓘ
work of Carl Friedrich Gauss on the arithmetic–geometric mean ⓘ |
| relatedConcept |
Borwein brothers’ AGM-based pi algorithms
ⓘ
Gauss transformation for elliptic integrals ⓘ Gauss–Legendre algorithm ⓘ
linked to:
arithmetic–geometric mean identities
Jacobi theta functions ⓘ Landen transformations ⓘ Ramanujan-type series for 1/\pi ⓘ modular invariants ⓘ |
| relatesQuantity |
Gamma function at rational arguments
ⓘ
arithmetic–geometric mean of two positive numbers ⓘ complete elliptic integral K(k) ⓘ complete elliptic integrals of the second kind ⓘ integrals of rational functions of trigonometric functions ⓘ modulus k of an elliptic integral ⓘ |
| usedFor |
derivation of modular equations
ⓘ
evaluation of special functions ⓘ fast algorithms for pi such as Gauss–Legendre algorithm ⓘ high-precision computation of mathematical constants ⓘ |
| usesConcept |
Beta function
ⓘ
Gamma function ⓘ arithmetic–geometric mean ⓘ complete elliptic integral of the first kind ⓘ elliptic functions ⓘ elliptic integrals ⓘ modular functions ⓘ pi ⓘ theta functions ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: arithmetic–geometric mean identities Description of subject: Arithmetic–geometric mean identities are a collection of formulas and relationships that express various mathematical constants and special functions in terms of the arithmetic–geometric mean of two numbers.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: arithmetic–geometric mean identities
linked to: arithmetic–geometric mean identities
linked to: arithmetic–geometric mean identities
linked to: arithmetic–geometric mean identities
linked to: arithmetic–geometric mean identities
linked to: arithmetic–geometric mean identities
linked to: arithmetic–geometric mean identities