# modular group

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##### 1: 21.5 Modular Transformations
βΊThe modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which $\xi(\boldsymbol{{\Gamma}})$ is determinate: …
##### 2: 23.1 Special Notation
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 $\mathbb{L}$ lattice in $\mathbb{C}$. … Cartesian product of groups $G$ and $H$, that is, the set of all pairs of elements $(g,h)$ with group operation $(g_{1},h_{1})+(g_{2},h_{2})=(g_{1}+g_{2},h_{1}+h_{2})$.
βΊThe main functions treated in this chapter are the Weierstrass $\wp$-function $\wp\left(z\right)=\wp\left(z|\mathbb{L}\right)=\wp\left(z;g_{2},g_{3}\right)$; the Weierstrass zeta function $\zeta\left(z\right)=\zeta\left(z|\mathbb{L}\right)=\zeta\left(z;g_{2},g_{3}\right)$; the Weierstrass sigma function $\sigma\left(z\right)=\sigma\left(z|\mathbb{L}\right)=\sigma\left(z;g_{2},g_{3}\right)$; the elliptic modular function $\lambda\left(\tau\right)$; Klein’s complete invariant $J\left(\tau\right)$; Dedekind’s eta function $\eta\left(\tau\right)$. …
##### 3: Ranjan Roy
βΊRoy has published many papers on differential equations, fluid mechanics, special functions, Fuchsian groups, and the history of mathematics. …He also authored another two advanced mathematics books: Sources in the development of mathematics (Roy, 2011), Elliptic and modular functions from Gauss to Dedekind to Hecke (Roy, 2017). …
##### 4: 23.20 Mathematical Applications
βΊFor conformal mappings via modular functions see Apostol (1990, §2.7). … βΊ
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##### 5: Software Index
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 Open Source With Book Commercial … 23 Weierstrass Elliptic and Modular Functions …
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• Open Source Collections and Systems.

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

• ##### 6: Mathematical Introduction
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 $(a,b]$ or $[a,b)$ half-closed intervals. … $m\equiv n\pmod{p}$ means $p$ divides $m-n$, where $m$, $n$, and $p$ are positive integers with $m>n$. …
βΊIn the Handbook this information is grouped at the section level and appears under the heading Sources in the References section. …
##### 7: Bibliography V
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• J. F. Van Diejen and V. P. Spiridonov (2001) Modular hypergeometric residue sums of elliptic Selberg integrals. Lett. Math. Phys. 58 (3), pp. 223–238.
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• N. Ja. Vilenkin and A. U. Klimyk (1991) Representation of Lie Groups and Special Functions. Volume 1: Simplest Lie Groups, Special Functions and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 72, Kluwer Academic Publishers Group, Dordrecht.
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• N. Ja. Vilenkin and A. U. Klimyk (1992) Representation of Lie Groups and Special Functions. Volume 3: Classical and Quantum Groups and Special Functions. Mathematics and its Applications (Soviet Series), Vol. 75, Kluwer Academic Publishers Group, Dordrecht.
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• N. Ja. Vilenkin and A. U. Klimyk (1993) Representation of Lie Groups and Special Functions. Volume 2: Class I Representations, Special Functions, and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht.
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• N. Ja. Vilenkin (1968) Special Functions and the Theory of Group Representations. American Mathematical Society, Providence, RI.
• ##### 8: Bibliography M
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• H. Maass (1971) Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.
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• I. D. Macdonald (1968) The Theory of Groups. Clarendon Press, Oxford.
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• N. W. Macfadyen and P. Winternitz (1971) Crossing symmetric expansions of physical scattering amplitudes: The $O(2,1)$ group and Lamé functions. J. Mathematical Phys. 12, pp. 281–293.
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• Magma (website) Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.
• ##### 9: Bibliography C
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• CAOP (website) Work Group of Computational Mathematics, University of Kassel, Germany.
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• G. Cornell, J. H. Silverman, and G. Stevens (Eds.) (1997) Modular Forms and Fermat’s Last Theorem. Springer-Verlag, New York.
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• J. E. Cremona (1997) Algorithms for Modular Elliptic Curves. 2nd edition, Cambridge University Press, Cambridge.
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• A. Cuyt, V. Petersen, B. Verdonk, H. Waadeland, W. B. Jones, and C. Bonan-Hamada (2007) Handbook of Continued Fractions for Special Functions. Kluwer Academic Publishers Group, Dordrecht.
• ##### 10: Bibliography
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• G. D. Anderson, S.-L. Qiu, M. K. Vamanamurthy, and M. Vuorinen (2000) Generalized elliptic integrals and modular equations. Pacific J. Math. 192 (1), pp. 1–37.
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• T. M. Apostol (1990) Modular Functions and Dirichlet Series in Number Theory. 2nd edition, Graduate Texts in Mathematics, Vol. 41, Springer-Verlag, New York.
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• V. I. Arnol’d (1972) Normal forms of functions near degenerate critical points, the Weyl groups ${A}_{k},{D}_{k},{E}_{k}$ and Lagrangian singularities. Funkcional. Anal. i PriloΕΎen. 6 (4), pp. 3–25 (Russian).
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• R. A. Askey, T. H. Koornwinder, and W. Schempp (Eds.) (1984) Special Functions: Group Theoretical Aspects and Applications. D. Reidel Publishing Co., Dordrecht.