Integral Form of Taylor Expansion and Its Generalizations to Discrete Calculus, Variational Inverse Problems, and the Duality of Differential and Integral Transforms

This paper systematically explores the integral remainder of Taylor's theorem and extends this idea to discrete calculus (differences and sums), to variational calculus (higher variations of functionals), and to inverse variational problems (recovering a functional from its Euler--Lagrange operator). We first give a rigorous proof of Taylor's theorem with integral remainder and discuss equivalent forms, connections with Lagrange remainder, error estimates, multivariate generalization, and the complex analytic counterpart. Next, we establish a discrete analogue: the Newton forward formula with a sum remainder, proved by discrete summation by parts and combinatorial identities; we also extend it to arbitrary step size. Then we introduce higher variations of functionals under Fr\'echet differentiability and derive the Taylor formula with integral remainder for functionals, illustrating it with the action functional. Subsequently, We derive the Helmholtz integrability conditions from the self-adjointness of the linearized operator, and present a rigorous proof of the path-integral construction of the Lagrangian based on a direct computation of the Euler--Lagrange operator. We then elevate this theory to the higher-order setting, providing a complete solution to the inverse problem for $k$-th order variations. Finally, we explore a continuous analogue of the Taylor series via the Mellin transform, which encodes the Taylor coefficients in the residues of its poles. These different manifestations reveal a unified theme: the reciprocity between differentiation (or its discrete/variational analogues) and integration (or summation/path integration), and the underlying duality between differential and integral transforms.