Research
Five analytical worlds, one structural language.
Rida's research connects successive differentiation, integral transforms, geometric representation, root structure, and finite matrix computation. Each area below states what is classical, what the book proposes, and where you can test it.
01
Higher-Order Parametric Derivatives via Definite Integrals
Evaluating high-order parametric derivatives as single definite integrals, without repeated quotient rules.
A study of one identity obtained from differentiation under the integral sign, which turns the n-th derivative of a boundary quotient into a single definite integral. Applied to trigonometric, exponential, and hyperbolic kernels, it gives phase-shifted formulas, Frullani-type evaluations via Fubini's theorem, and a route to the Laplace transform.
02
An Operator-Based Framework for Integral Transforms
Laplace, Fourier, and Mellin transforms generated from one kernel by repeated differentiation.
A framework that builds integral transforms from the Maclaurin coefficients of the input, letting each coefficient select a derivative of the kernel 1/s. The construction reproduces the Laplace transform, leads to Fourier and Mellin-type transforms through the choice of rank, and extends to general kernels, convolution, and fractional integration.
03
Mathematical Geometry: Angular–Hyperbolic Representation
Reading Laplace transforms through circular and hyperbolic coordinates in the damping–frequency plane.
A geometric reading of the Laplace transform in which damping and frequency are separate coordinates of a point in the plane. Bounded oscillatory kernels take a polar form governed by an angle, unbounded kernels a hyperbolic form, and composite kernels such as sinc and Bessel functions are treated as angularly saturated.
04
Root Geometry and Higher Derivatives of Rational Functions
Closed formulas for derivatives of rational functions, read from the geometry of roots.
Closed formulas for the n-th derivative of rational functions, obtained by writing each quadratic factor through its roots. Complex-conjugate roots lead to a trigonometric law and real roots to a hyperbolic law, and partial fractions extend the method to linear numerators, cubic denominators, and Taylor coefficients.
05
Matrix Boundary Method
Truncated Laplace integrals and linear ODEs reduced to one resolvent linear system.
A finite-dimensional method for integrals of the form e^(bx) f(x) over [0, a], where f lies in a function space closed under differentiation. Differentiation becomes a constant matrix, and the integral reduces to one linear system and a boundary evaluation, with extensions to products, matrix exponentials, the Laplace transform, and first-order linear ODEs.
- Derivatives
- Operators
- Transforms
- Geometry
- Root structure
- Matrices
Scientific classification
Every statement says what kind of statement it is.
- Classical Foundation
- Established mathematics found in standard textbooks and references.
- Published Book Content
- A statement as it appears in Rida Abu-Sokon's published book.
- Book Framework
- The organising framework, notation, or terminology introduced in the book.
- Proposed Formulation
- A formulation proposed in the book that goes beyond the classical presentation.
- Computational Demonstration
- A numerical or visual demonstration. It illustrates; it does not prove.
- Exploratory Note
- Work in progress or an open question, stated as such.
- Educational Interpretation
- RIDA MATH's explanation of mathematics for learning purposes.