General Mathematics

    

The Leaf Theorem

Authors: Marciano L. Legarde

I present two results known as the Leaf Theorems, that were initially noticed via numerical experiment and subsequently proved analytically. Each of these theorems illustrates that the disparity between rapidly oscillating and slow growth functions, and rapidly diminishing functions and disappearing power functions, respectively, result in constant, interpretable, and finite values when integrated over the unit segment. Together, these results demonstrate that contrasting mathematical behaviors may cancel in the process of integrating these functions and result in interpretable and finite quantities, and offer apparent and pedagogical demonstrations of real analysis convergence. They could prove helpful for pedagogy, for use in asymptotic analysis, and for applications in number and numerical methods and in probability, and could serve to inform and educate analysts and students in these and related fields.

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[v1] 2025-10-05 23:49:44

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