Rotational Integral Transforms of the Trigonometric Kernel Family: Structural Classification and Correspondence from Trigonometric Functions to L-Functions

This paper establishes and systematically studies a class of rotational integral transforms\[ V[f](s)=i^s/Γ(s) M[f(it)](s),\]namely, the composite of a Wick rotation, the Mellin transform, and \(Γ(s)\)-normalization. The central starting point of the paper is that this composite uniformly maps periodic-sequence kernels and Ramanujan Master Theorem-type kernels to the form “a pure phase factor \(i^s\) multiplied by an analytic object free of Gamma factors,” namely\[ V[f_a](s)=i^sD_a(s), V[f](s)=i^sφ(-s).\]The main objects of the paper are the trigonometric kernel family \( x, x, x,tan x\) together with their powers, products, and combinations. We give their explicit structural classification and correspondence to \(ζ(s),β(s),η(s)\), and general Dirichlet series. At the same time, we establish operational rules such as multiplication by \(x^m\), differentiation, multiplication by \(log x\), phase weighting, variable translation, and pointwise multiplication of sequences, and we systematically treat Hadamard finite parts and strips of convergence. The paper further translates the classical Ramanujan Master Theorem, the Bradshaw–Vignat operator identity, and Atale’s Mellin functional equation for modular forms into the \( V\)-framework. It explicitly points out that \( V\) is suited to representing cases in which the right-hand side is an \(L\)-function value, whereas the Gamma-retaining variant\[{ V}[f](s)=Γ(s) V[f](s)\]is suited to representing cases whose functional equation is a pure-phase relation. Chapter 5 discusses the multivariate \( V_2\), provides new entries such as double integral representations for \(β(s)β(r)\) and \(ζ(s)ζ(r)\), and clarifies the structural boundaries of the framework. All 76 boxed formulas in the paper have been verified numerically on both sides using mpmath, with tolerance \(10⁻¹⁰\) and a maximum residual of approximately \(1.1×10⁻¹¹\). The contributions of the paper lie mainly in the dictionary of trigonometric kernels, the identification of the division of labor between \( V\) and \({ V}\), the unified phase structure, and verifiable new integral representations.