椭圆函数及其应用（第一卷）：函数论和分析基础（英文版）The Elliptic Functions and Their Applications, First

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目录
- 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

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