Theory of Analytic Functions

The "Theory of Analytic Functions" is a mathematical concept primarily developed by Joseph-Louis Lagrange in the late 18th century. Lagrange sought to establish the calculus on a purely algebraic foundation, moving away from geometric concepts and infinitesimals.12

In his 1797 treatise, Théorie des fonctions analytiques, Lagrange defined the derivative of a function f(x) as the coefficient of i in the Taylor expansion of f(x + i). He aimed to develop a comprehensive theory of differential and integral calculus using this algebraic approach, though later mathematicians like Cauchy and Riemann established the calculus on an arithmetic basis using limits. The concept of analytic functions was later extended to complex numbers, where differentiability implies the existence of all derivatives and a power series expansion.134