1 Introduction 1 2 Riemann Surface 3 2.1 Introduction_z^(1/2) as a single-valued function 3 2.2 Geometric construct the Riemann Surface 5 2.3 Geometric and algebraic structure of Riemann surface of genus N 7 2.4 The a , b-cycles 13 2.5 Path integrals on Riemann surface of genus N 16 2.6 Evaluations of path integrals on Riemann surfaceof genus N with horizontal cut structure 19 2.7 Evaluations of path integrals on Riemann surface of genus N with vertical cut structure 31 2.8 Evaluations of path integrals on Riemann surface of genus N with slat cut structure 39 3 Elliptic function 47 3.1 General definitions and properties 47 3.1.1 Doubly-periodic function 47 3.1.2 Elliptic function 47 3.1.3 Period-parallelograms 47 3.1.4 Simple properties of elliptic functions 48 3.2 Weierstrass elliptic function 49 3.2.1 Definition 49 3.2.2 The function 50 3.3 The Theta-functions 52 3.3.1 Definition 52 3.3.2 The four types of Theta-functions 53 3.4 The Jacobian elliptic functions 55 3.4.1 The Jacobian elliptic functions, sn u, cn u, dn u 55 3.4.2 Glaisher’s notation for quotients 56 3.4.3 The addition-theorem for the function sn u, cn u, dn u 57 3.4.4 The constant K, K’ 57 3.4.5 The periodic properties (associated with K) of the Jacobian elliptic functions 57 3.4.6 The periodic properties (associated with K+iK’ ) of the Jacobian elliptic functions 58 3.4.7 The periodic properties (associated with iK’) of the Jacobian elliptic functions 59 4 The Jacobian elliptic functions and the sine-Gordon equation 60 5 Conclusion 67 Appendices 68 References 69