References

1) Official citation (DOI + BibTeX)

Use the version DOI when citing results tied to a specific release, and the concept DOI when citing the project as a whole.

@software{gossa_hierarchical_calculus_2025,
  author = {Gossa Ahmed},
  title  = {Hierarchical Calculus},
  year   = {2025},
  doi    = {10.5281/zenodo.17917302},
  url    = {https://github.com/GOSSAAHMED/gossa-math}
}

GOSSA AHMED. Hierarchical Calculus — Official Website. Zenodo. https://doi.org/10.5281/zenodo.17917302 (2025).

2) Standard mathematical references

• Walter Rudin — Principles of Mathematical Analysis.
• Terence Tao — Analysis I / II.
• Tom M. Apostol — Calculus, Vol. 1 & 2.
• Boyce & DiPrima — Elementary Differential Equations and Boundary Value Problems.
• Coddington & Levinson — Theory of Ordinary Differential Equations.
• NIST — Digital Library of Mathematical Functions (DLMF).

3) Numerical analysis & approximation

• Burden & Faires — Numerical Analysis.
• Sauer — Numerical Analysis.
• Nicholas J. Higham — Accuracy and Stability of Numerical Algorithms.
• Lloyd N. Trefethen — Approximation Theory and Approximation Practice.

4) Physics references (for applied examples)

• Zemansky & Dittman — Heat and Thermodynamics.
• H. B. Callen — Thermodynamics and an Introduction to Thermostatistics.
• Goldstein — Classical Mechanics.
• Halliday, Resnick & Walker — Fundamentals of Physics.

5) Domain assumptions (important for rigor)

Increasing rank may shrink the domain due to iterated logarithms. A compact summary:

- Rank 0: no special domain constraints beyond classical differentiability.
- Rank 1: typically requires x>0 and f(x)>0 when using ln x and ln f.
- Rank 2: requires x>1 with ln(x)>0 and f(x)>1 with ln(f)>0.
- Rank 3+: adds further conditions for iterated logarithms and avoids singularities.

6) Project links

• Website (AR): Arabic
• Website (EN): English
• GitHub: Repository
• Zenodo (concept): 10.5281/zenodo.17917302