General addition formula for meromorphic functions derived from residue theorem

A meromorphic function is characterized by its singularity, and residues of the poles give us various information regarding the function. In the paper the algebraic relations of residues on some meromorphic functions are considered and a generic form of addition formula is induced from them. As the applications of that formula, the addition formula for rational function, hypergeometric function, elliptic function, and Riemann's Zeta function are studied. Also the general multiplication formula is discussed based on the previous discourse.