前辅文
Introduction Compilation of theorems on analytic functions
§1. Concept of the analytic function of a complex variable
§2. Of the integrals of the analytic functions
§3. Explanation of analytic functions by power series
§4. The Cauchy integral formula and the Cauchy-Taylor series
§5. The analytic continuation and the additions to the visual aids caused by it
§6. The field F of an analytic function and its singular points
§7. Of the mappings mediated by analytic functions
§8. Concept of the residue and theorems about residues
§9. Laurent’s series and theorem. Conclusions about unique functions
§10. Entire rational functions and their inverse functions
§11. Entire transcendental functions. Exponential function and logarithm
§12. Product representation of the entire transcendental functions
§13. The rational functions and their inverse functions
§14. The linear substitutions and the concept of circular relationship
§15. The linear substitutions of the second kind and the indirect circular relationships
§16. General information about algebraic functions and structures
§17. The degree of connectivity of a Riemann surface Fm
§18.More about algebraic functions especially at p=0 and p=1. Basic problem of the theory of elliptic functions
§19. Solution of second-order linear homogeneous differential equations
§20. Comments on the hypergeometric differential equation
Section one Basics of the theory of elliptic functions of the first level
Chapter one The elliptic integrals and their normal forms belonging to the first level
§1. The branch form, its invariants and its normal formof the first level
§2. Excursus on linear substitutions and their groups of finite order
§3. The linear transformations of the branch forminto itself
§4. Invariance of J under any rational transformation of the Riemann surface F2
§5. General remarks on the elliptic integrals
§6. The three kinds of elliptic integrals and the elementary integrals
§7. The normal forms of the integrals of the third kind belonging to the first level
§8. The periods of the elliptic integrals and the relations existing between them
§9. The transcendentally normalized integrals of the second and third kind
Chapter two The first-level elliptic integral of the first kind and the mappings mediated by it
§1. The field F∞ of the function u(z) and its mapping onto the u-plane using special cross-sections
§2. The periods ω1, ω2 and the period quotient ω of the reduced cross-section system
§3. Transition to any cross-section system and linear transformation of the periods
§4. Behavior of the integral of the first kind for unique transformations of the surface F2
Chapter three The elliptic functions of the first level
§1. The integral of the first kind u as a “uniformizing” variable of the Riemann surface F2
§2. The field of elliptic functions, the special functions φ(u), φ'(u) and the normal integral ζ(u)
§3. Power series for the functions φ(u), φ'(u) and ζ(u)
§4. Representation of the elementary integrals in u. Preliminary information about the addition theorems
§5. Representation of all elliptic functions of the field by the functions ζ, φ, φ', φ'',
§6. The entire transcendental function σ(u; g2, g3)
§7. Representation of the elliptic functions by the σ-function
§8. The elliptic functions of the second and third kind
§9. Number theorem about elliptic functions with given poles together with consequences
Chapter four The unique double-periodic functions of the first level
§1. The substitution group Γ(u) of the double-periodic functions
§2. The discontinuity region of the substitution group Γ(u)
§3. Introduction of a hexagonal discontinuity region of Γ(u) and explanation of a reduced pair of periods
§4. Of the transformations of the group Γ(u) into itself
§5. Concept of double-periodic functions and residue theorems
§6. On the convergence of certain double series
§7. Proof of the existence of double-periodic functions
§8. Partial fraction series for the functions φ(u), φ'(u) and ζ(u) and their consequences.
§9. The functions of the annular region together with applications
§10. The system of all elliptic functions and the degeneration of them
Chapter five The elliptic modular functions of the first level and their inverse functions
§1. The modular group Γ(u) and its extension by a reflection
§2. The triangular network of the ω-half-plane and the discontinuity region of the modular group
§3. The generating substitutions of the modular group
§4. The elliptic modular functions of the first level
§5. The elliptic modular forms of the first level
§6. The periods η1, η2 as functions of ω1, ω2. Product expansion of the discriminant
§7. Differentiation processes for the production of modular forms
§8. The double-periodic functions of the first level as functions of three arguments
§9. Differential equations of the periods with respect to the invariants. Inversion of the modular functions
§10. The normalized periods as hypergeometric functions of J
Section two Basics of the theory of elliptic functions of the second level
Chapter one The normal forms of the second and fourth level of the branch form and the elliptic integrals
§1. The simplest irrational invariants of the branch form
§2. Relation of the irrational invariants of the branch formto the rational invariants
§3. The normal formof the second level of the branch form
§4. The normal formof the fourth level of the branch form
§5. The normal forms of the second and fourth levels of the elliptic integrals
§6. The Legendre’s normal integrals
Chapter two The elliptic functions of the second level
§1. The principle of level division and the concept of the elliptic function of the n-th level
§2. The congruence groups of the second level in the Γ(u)
§3. The functions of the second level
§4. The Jacobian functions sn w, cn w, dn w
§5. The derivatives of the functions sn w, cn w, dn w and power series of the same
§6. Representation of the functions sn w, cn w, dn w as quotients of entire transcendental functions
§7. Product developments of the elliptic functions of the second level
§8. Laurent and Fourier series for the functions sn, cn and dn
§9. The Fourier series for the entire functions σ1(u), σ2(u), σ3(u) and the theta functions
§10. The theta functions with arbitrary characteristic
§11. The higher order theta functions with arbitrary characteristic
§12. The theta functions of the m-th order as entire elliptic functions of the third kind
Chapter three The modular functions of the second level and the linear transformation of the elliptic functions of the second level
§1. The discontinuity region of the principal congruence group of the second level
§2. The elliptic modular functions of the second level
§3. The elliptic modular forms of the second level
§4. Modular functions of higher levels in the theory of elliptic functions
§5. The periods considered as functions of the integral modulus
§6. The elliptic functions of the second level as functions of two arguments. Degenerations
§7. Behavior of the elliptic functions of the second level with any period substitution
§8. Behavior of the theta functions with any period substitutions
§9. General law about the behaviour of the theta functions with period substitutions
§10. Application to the theory of Gaussian sums
Subject index
