J-invariant (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "J-invariant" in English language version.

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Petersson, Hans (1932). "Über die Entwicklungskoeffizienten der automorphen Formen". Acta Mathematica. 58 (1): 169–215. doi:10.1007/BF02547776. MR 1555346.
Rademacher, Hans (1938). "The Fourier coefficients of the modular invariant j(τ)". American Journal of Mathematics. 60 (2): 501–512. doi:10.2307/2371313. JSTOR 2371313. MR 1507331.

arxiv.org (Global: 69^th place; English: 59^th place)

Milne, Steven C. (2000). "Hankel Determinants of Eisenstein Series". arXiv:math/0009130v3. The paper uses a non-equivalent definition of $\Delta$ , but this has been accounted for in this article.

books.google.com (Global: 3^rd place; English: 3^rd place)

Gareth A. Jones and David Singerman. (1987) Complex functions: an algebraic and geometric viewpoint. Cambridge UP. [1]

doi.org (Global: 2^nd place; English: 2^nd place)

Petersson, Hans (1932). "Über die Entwicklungskoeffizienten der automorphen Formen". Acta Mathematica. 58 (1): 169–215. doi:10.1007/BF02547776. MR 1555346.
Rademacher, Hans (1938). "The Fourier coefficients of the modular invariant j(τ)". American Journal of Mathematics. 60 (2): 501–512. doi:10.2307/2371313. JSTOR 2371313. MR 1507331.
Cummins, Chris J. (2004). "Congruence subgroups of groups commensurable with PSL(2,Z)$ of genus 0 and 1". Experimental Mathematics. 13 (3): 361–382. doi:10.1080/10586458.2004.10504547. ISSN 1058-6458. S2CID 10319627. Zbl 1099.11022.
Chudnovsky, David V.; Chudnovsky, Gregory V. (1989), "The Computation of Classical Constants", Proceedings of the National Academy of Sciences of the United States of America, 86 (21): 8178–8182, Bibcode:1989PNAS...86.8178C, doi:10.1073/pnas.86.21.8178, ISSN 0027-8424, JSTOR 34831, PMC 298242, PMID 16594075.

harvard.edu (Global: 18^th place; English: 17^th place)

ui.adsabs.harvard.edu

Chudnovsky, David V.; Chudnovsky, Gregory V. (1989), "The Computation of Classical Constants", Proceedings of the National Academy of Sciences of the United States of America, 86 (21): 8178–8182, Bibcode:1989PNAS...86.8178C, doi:10.1073/pnas.86.21.8178, ISSN 0027-8424, JSTOR 34831, PMC 298242, PMID 16594075.

jstor.org (Global: 26^th place; English: 20^th place)

Rademacher, Hans (1938). "The Fourier coefficients of the modular invariant j(τ)". American Journal of Mathematics. 60 (2): 501–512. doi:10.2307/2371313. JSTOR 2371313. MR 1507331.
Chudnovsky, David V.; Chudnovsky, Gregory V. (1989), "The Computation of Classical Constants", Proceedings of the National Academy of Sciences of the United States of America, 86 (21): 8178–8182, Bibcode:1989PNAS...86.8178C, doi:10.1073/pnas.86.21.8178, ISSN 0027-8424, JSTOR 34831, PMC 298242, PMID 16594075.

nih.gov (Global: 4^th place; English: 4^th place)

ncbi.nlm.nih.gov

Chudnovsky, David V.; Chudnovsky, Gregory V. (1989), "The Computation of Classical Constants", Proceedings of the National Academy of Sciences of the United States of America, 86 (21): 8178–8182, Bibcode:1989PNAS...86.8178C, doi:10.1073/pnas.86.21.8178, ISSN 0027-8424, JSTOR 34831, PMC 298242, PMID 16594075.

pubmed.ncbi.nlm.nih.gov

Chudnovsky, David V.; Chudnovsky, Gregory V. (1989), "The Computation of Classical Constants", Proceedings of the National Academy of Sciences of the United States of America, 86 (21): 8178–8182, Bibcode:1989PNAS...86.8178C, doi:10.1073/pnas.86.21.8178, ISSN 0027-8424, JSTOR 34831, PMC 298242, PMID 16594075.

projecteuclid.org (Global: 3,707^th place; English: 2,409^th place)

Cummins, Chris J. (2004). "Congruence subgroups of groups commensurable with PSL(2,Z)$ of genus 0 and 1". Experimental Mathematics. 13 (3): 361–382. doi:10.1080/10586458.2004.10504547. ISSN 1058-6458. S2CID 10319627. Zbl 1099.11022.

researchgate.net (Global: 120^th place; English: 125^th place)

The equality holds if the arithmetic–geometric mean $\operatorname {M} (a,b)$ of complex numbers $a,b$ (such that $a,b\neq 0;a\neq \pm b$ ) is defined as follows: Let $a_{0}=a$ , $b_{0}=b$ , $a_{n+1}=(a_{n}+b_{n})/2$ , $b_{n+1}=\pm {\sqrt {a_{n}b_{n}}}$ where the signs are chosen such that $|a_{n}-b_{n}|\leq |a_{n}+b_{n}|$ for all $n\in \mathbb {N}$ . If $|a_{n}-b_{n}|=|a_{n}+b_{n}|$ , the sign is chosen such that $\Im (b_{n}/a_{n})>0$ . Then $\operatorname {M} (a,b)=\lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}$ . When $a,b$ are positive real (with $a\neq b$ ), this definition coincides with the usual definition of the arithmetic–geometric mean for positive real numbers. See The Arithmetic-Geometric Mean of Gauss by David A. Cox.