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 ftmction 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 circularrelation-ships
  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 form of the first level
  2. Excursus on linear substitutions and their groups of finite order
  3. The linear transformations of the branch form into itself
  4. Invariance of] 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 me-diated by it
  1. The field Fco of the function u(z) and its mapping onto the u-plane using special cross-sections
  2. The periods to1, to2 and the period quotient to of the reduced cross-section system
  3. Transition to any cross-section system and linear transformation of the pe- riods
  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 ftmctions, the special functions □a(u), □'(u) and the normal integral ((u)
  3. Power series for the functions □(u), □'(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, □, □',
  6. The entire transcendental function σ(u; g2, g3)
  7. Representation of the elliptic functions by the nr-fimction
  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 F(u) of the double-periodic functions
  2. The discontinuity region of the substitution group F(u)
  3. Introduction of a hexagonal discontinuity region of F(u) and explanation of a reduced pair of periods
Chapter five
  The elliptic modular functions of the first level and their inverse functions
  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
Chapter two
  The elliptic functions of the second level
Chapter three
  The modular functions of the second level and the linear transfornia-
  tion of the elliptic functions of the second level