Elliptic and modular functions from Gauss to Dedekind to Hecke (Q2833167)

This book covers Elliptic and Modular functions and their applications. The book consists of 16 chapters. The subject and scope of the book is the potential to contribute to the field of physics and engineering besides other fields of mathematics, primarily Analytic Number theory. The Contents of this book are given in detail as follows:NEWLINENEWLINENEWLINE1 -- The Basic Modular Forms of the Nineteenth CenturyNEWLINENEWLINENEWLINEThe modular group and some of its subgroups were used especially in the theory of the Modular Forms and functions, in the theory of theta functions and the other areas. This section includes the Modular Group and Modular Forms.NEWLINENEWLINENEWLINE2 -- Gauss's Contributions to Modular formsNEWLINENEWLINENEWLINEThis section includes the following subsections: 2.1 Early Work on Elliptic Integrals 2.2 Landen and Legendre's Quadratic Transformation 2.3 Lagrange's Arithmetic-Geometric Mean 2.4 Gauss on the Arithmetic-Geometric Mean 2.5 Gauss on Elliptic Functions 2.6 Gauss: Theta Functions and Modular Forms and 2.7 Exercises including related subjects of this section.NEWLINENEWLINENEWLINE3 -- Abel and Jacobi on Elliptic FunctionsNEWLINENEWLINENEWLINEIn the late 1790s Gauss discovered elliptic functions and their double periodicity. Abel died in 1829 at the age of twenty-six of tuberculosis. In the short period between his first paper and his death, he published seven more papers on elliptic functions, including another paper of one hundred pages, ``Précis d'une théorie des fonctions elliptiques.'' In the spring of 1827, Jacobi started work on elliptic integrals under the influence of Legendre's 1817 work on elliptic integrals. Jacobi made very significant contributions to the theory elliptic functions, the theory of hypergeometric functions and orthogonal polynomials, functional determinants and the change of variables formula for \(n\)-dimensional integrals, Hamiltonian dynamics, partial differential equations, number theory, and astronomy.NEWLINENEWLINEThis section includes the following subsections: 3.2 Jacobi on Transformations of Orders 3 and 5, 3.3 The Jacobi Elliptic Functions 3.4 Transformations of Order n and Infinite Products 3.5 Jacobi's Transformation Formulas 3.6 Equivalent Forms of the Transformation Formulas 3.7 The First and Second Transformations 3.8 Complementary Transformations 3.9 Jacobi's First Supplementary Transformation 3.10 Jacobi's Infinite Products for Elliptic Functions 3.11 Jacobi's Theory of Theta Functions 3.12 Jacobi's Triple Product Identity 3.13 Modular Equations and Transformation Theory 3.14 Exercises including related subjects of this section.NEWLINENEWLINENEWLINE4 -- Eisenstein and Hurwitz Export citation.NEWLINENEWLINENEWLINEThis section includes the following subsections: 4.2 Eisenstein's Theory of Trigonometric Functions 4.3 Eisenstein's Derivation of the Addition Formula 4.4 Eisenstein's Theory of Elliptic Functions 4.5 Differential Equations for Elliptic Functions 4.6 The Addition Theorem for the Elliptic Function 4.7 Eisenstein's Double Product 4.8 Elliptic Functions in Terms of the \(\varphi\) Function 4.9 Connection of \(\varphi\) with Theta Functions 4.10 Hurwitz's Fourier Series for Modular Forms 4.11 Hurwitz's Proof That \((\omega)\) Is a Modular Form 4.12 Hurwitz's Proof of Eisenstein's Result 4.13 Kronecker's Proof of Eisenstein's Result 4.14 Exercises including related subjects of this section.NEWLINENEWLINENEWLINE5 -- Hermite's Transformation of Theta FunctionsNEWLINENEWLINENEWLINEHermite also made important contributions to the theory of quadratic forms in two or more variables, the theory of theta functions and their generalizations, the theory of elliptic and abelian functions, and several topics in analysis. This section includes the following subsections: 5.2 Hermite's Proof of the Transformation Formula 5.3 Smith on Jacobi's Formula for the Product of Four Theta Functions 5.4 Exercises including related subjects of this section.NEWLINENEWLINENEWLINE6 -- Complex Variables and Elliptic unctionsNEWLINENEWLINENEWLINEIn the 1830s, Jacobi created a basis for the theory of elliptic functions by defining an elliptic function as the ratio of two theta functions. In the 1840s, Joseph Liouville (1809--1882) gave a different direction to the theory of elliptic functions, defining them as doubly periodic analytic functions and then deriving their properties on the basis of this definition. In 1847, Liouville's original approach to elliptic functions as doubly periodic meromorphic functions opened the door to the application of Cauchy's theory of residues to this class of functions. In 1848, Hermite made use of Cauchy's theory to show that the sum of the residues of an elliptic function was zero inside a period parallelogram.NEWLINENEWLINEThis section includes the following subsections: 6.1 Historical Remarks on the Roots of Unity 6.2 Simpson and the Ladies Diary 6.3 Development of Complex Variables Theory 6.4 Hermite: Complex Analysis in Elliptic Functions 6.5 Riemann: Meaning of the Elliptic Integral 6.6 Weierstrass's RigorizationNEWLINENEWLINENEWLINE7 -- Hypergeometric FunctionsNEWLINENEWLINENEWLINEThis section includes the following subsections: 7.1 Preliminary Remarks 7.2 Stirling 7.3 Euler and the Hypergeometric Equation 7.4 Pfaff's Transformation 7.5 Gauss and Quadratic Transformations 7.6 Kummer on the Hypergeometric Equation 7.7 Riemann and the Schwarzian Derivative 7.8 Riemann and the Triangle Functions 7.9 The Ratio of the Periods \(K_K\) as a Conformal Map 7.10 Schwarz: Hypergeometric Equation with Algebraic Solutions 7.11 Exercises including related subjects of this section.NEWLINENEWLINENEWLINE8 -- Dedekind's Paper on Modular unctionsNEWLINENEWLINENEWLINEThis section includes the following subsections: 8.2 Dedekind's Approach 8.3 The Fundamental Domain for \(\mathrm{SL}_2(\mathbb Z)\) 8.4 Tesselation of the Upper Half-plane 8.5 Dedekind's Valency Function 8.6 Branch Points 8.7 Differential Equations 8.8 Dedekind's eta Function 8.9 The Uniqueness of \(k^2\) 8.10 The Connection of Dedekind eta function with Theta Functions 8.11 Hurwitz's Infinite Product for Dedekind eta function 8.12 Algebraic Relations among Modular Forms 8.13 The Modular Equation 8.14 Singular Moduli and Quadratic Forms 8.15 Exercises including related subjects of this section.NEWLINENEWLINENEWLINE9 -- The Dedekind eta Function and Dedekind SumsNEWLINENEWLINENEWLINEDedekind sums occur in several contexts other than the study of the Dedekind eta function and their modular equations. Dedekind sums also occur in in counting the lattice points inside certain triangles, parallelepipeds, pyramids, and tetrahedrons. There are various interesting and important generalization of the Dedekind sums.NEWLINENEWLINENEWLINEThis section includes the following subsections: 9.2 Riemann's Notes 9.3 Dedekind Sums in Terms of a Periodic Function 9.4 Rademacher 9.5 Exercises including related subjects of this section.NEWLINENEWLINENEWLINE10 -- Modular Forms and Invariant TheoryNEWLINENEWLINENEWLINEInvariant theory played an important role in the development of modular functions; tracing the beginnings of invariant theory from 1840 to 1855 shows how George Boole defined an invariant and gives insight into the methods used by Boole, Arthur Cayley, and J. J. Sylvester to compute basic invariants, including those used in modular function theory. This section includes the following subsections: 10.2 The Early Theory of Invariants 10.3 Cayley's Proof of a Result of Abel 10.4 Reduction of an Elliptic Integral to Riemann's Normal Form 10.5 The Weierstrass Normal Form 10.6 Proof of the Infinite Product for delta 10.7 The Multiplier in Terms of square root of delta.NEWLINENEWLINENEWLINE11 -- The Modular and Multiplier EquationsNEWLINENEWLINENEWLINEThis section includes the following subsections: Jacobi's Multiplier Equation 11.3 Sohnke's Paper on Modular Equations 11.4 Brioschi on Jacobi's Multiplier Equation 11.5 Joubert on the Multiplier Equation 11.6 Kiepert and Klein on the Multiplier Equation 11.7 Hurwitz: Roots of the Multiplier Equation 11.8 Exercises including related subjects of this section.NEWLINENEWLINENEWLINE12 -- The Theory of Modular Forms as Reworked by HurwitzNEWLINENEWLINENEWLINEThis section includes the following subsections: The Fundamental Domain 12.3 An Infinite Product as a Modular Form 12.4 The J-Function 12.5 An Application to the Theory of Elliptic Functions.NEWLINENEWLINENEWLINE13 -- Ramanujan's Euler Products and Modular FormsNEWLINENEWLINENEWLINEThis section includes the following subsections: 13.2 Ramanujan's tau Function 13.3 Ramanujan: Product Formula 13.5 The Arithmetic Function Ramanujan tau function 13.6 Mordell on Euler Products 13.7 Exercises including related subjects of this section.NEWLINENEWLINENEWLINE14 -- Dirichlet Series and Modular FormsNEWLINENEWLINENEWLINEThis section includes the following subsections: 14.2 Functional Equations for Dirichlet Series 14.3 Theta Series in Two Variables 14.4 Exercises including related subjects of this section.NEWLINENEWLINENEWLINE15 -- Sums of SquaresNEWLINENEWLINENEWLINEThis section includes the following subsections: 15.2 Jacobi's Elliptic Functions Approach 15.3 Glaisher 15.4 Ramanjuan's Arithmetical Functions 15.5 Mordell: Spaces of Modular Forms 15.6 Hardy's Singular Series 15.7 Hecke's Solution to the Sums of Squares Problem 15.8 Exercises including related subjects of this section.NEWLINENEWLINENEWLINE16 -- The Hecke OperatorsNEWLINENEWLINENEWLINEErich Hecke defined an infinite sequence of operators on the finite dimensional vector of homogeneous modular forms of weight \(k\). He first worked out the theory for level 1 functions, that is, modular forms for the full modular group. This section includes the following subsections: 16.2 The Hecke Operators \(T(n)\) 16.3 The Operators \(T(n)\) in Terms of Matrices \(\lambda(n)\) 16.4 Euler Products 16.5 Eigenfunctions of the Hecke Operators 16.6 The Petersson Inner Product 16.7 Exercises including related subjects of this section.NEWLINENEWLINENEWLINEAppendix: Translation of Hurwitz's paper of 1904: 1. Equivalent Quantities 2. The Modular Forms \(G_n(\omega 1, \omega 2)\) 3. The Representation of the Function \(G_n\) by Power Series 4. The Modular Form \(\Delta(\omega_1, \omega_2)\) 5. The Modular Function \(J(\omega)\) 6. Applications to the Theory of Elliptic Functions