This chapter presents the classical example of entire and meromorphic functions of order 2, illustrating most of the theorems discussed in the book. A self-contained analytic continuation of the topics is covered in Chapters 3, 4, and 18 of the author's book on Elliptic Functions [La 73]. Section X IV, §1 discusses the Liouville theorems. It defines two complex numbers ω₁ and ω₂ that are linearly independent over the real numbers. The lattice generated by ω₁ and ω₂ consists of all complex numbers of the form mω₁ + nω₂, where m and n are integers. This lattice is closed under addition and multiplication by integers, making it a subgroup of C. Since C is a 2-dimensional vector space over R, ω₁ and ω₂ form a basis of C over R. Congruence modulo L is defined as z ≡ w mod L if z - w ∈ L. This relation is an equivalence relation, meaning it is reflexive, symmetric, and transitive. The set of equivalence classes mod L is denoted by C/L. If z ≡ w mod L and n is an integer, then nz ≡ nw mod L. If λ is a non-zero complex number, then λL is also a lattice, and λz ≡ λw mod λL. A fundamental parallelogram for the lattice is defined as the set of points α + t₁ω₁ + t₂ω₂, where 0 ≤ tᵢ ≤ 1. This parallelogram can also be defined with 0 ≤ tᵢ < 1.This chapter presents the classical example of entire and meromorphic functions of order 2, illustrating most of the theorems discussed in the book. A self-contained analytic continuation of the topics is covered in Chapters 3, 4, and 18 of the author's book on Elliptic Functions [La 73]. Section X IV, §1 discusses the Liouville theorems. It defines two complex numbers ω₁ and ω₂ that are linearly independent over the real numbers. The lattice generated by ω₁ and ω₂ consists of all complex numbers of the form mω₁ + nω₂, where m and n are integers. This lattice is closed under addition and multiplication by integers, making it a subgroup of C. Since C is a 2-dimensional vector space over R, ω₁ and ω₂ form a basis of C over R. Congruence modulo L is defined as z ≡ w mod L if z - w ∈ L. This relation is an equivalence relation, meaning it is reflexive, symmetric, and transitive. The set of equivalence classes mod L is denoted by C/L. If z ≡ w mod L and n is an integer, then nz ≡ nw mod L. If λ is a non-zero complex number, then λL is also a lattice, and λz ≡ λw mod λL. A fundamental parallelogram for the lattice is defined as the set of points α + t₁ω₁ + t₂ω₂, where 0 ≤ tᵢ ≤ 1. This parallelogram can also be defined with 0 ≤ tᵢ < 1.