Euler’s number e is one of the central constants in mathematics, appearing in analysis, differential equations, probability, and complex variables. In this paper, we examine the identity
e = 4th root of 54 + [((-1)^(-i/π) − 4th root of 54) / Σ from k = 1 to ∞ of √5 / (πk)^(5k)] × Σ from k = 1 to ∞ of √5 / (πk)^(5k)
The expression combines a complex power, a rapidly convergent infinite series, and a fourth root of an integer. Using the principal branch of the complex logarithm, the term (-1)^(-i/π) is evaluated exactly as e. The infinite series
Σ from k = 1 to ∞ of √5 / (πk)^(5k)
is absolutely convergent and decays extremely fast because the exponent grows with k. After substituting the complex term and simplifying algebraically, the expression reduces to
4th root of 54 + (e − 4th root of 54) = e
Thus, the identity is verified exactly. Although the formula does not define a new constant or provide a new representation of e, it serves as an instructive example of how complex exponentiation and infinite series can be combined into a compact symbolic identity. The paper highlights the role of branch selection in complex analysis, the behavior of rapidly convergent series, and the algebraic cancellation that leads to the final result.
Table of Contents
Introduction
Methodology
Evaluation of the complex power
Study of the infinite series
Algebraic simplification
Convergence Analysis
Numerical Verification
Discussion
Conclusion
Objectives and Topics
The primary objective of this paper is to analyze a constructed mathematical identity involving Euler's number (e), a complex exponent, and a rapidly convergent infinite series, demonstrating how seemingly complex expressions simplify through rigorous interpretation.
- Complex exponentiation and the role of the principal logarithm branch.
- Analysis of super-exponentially decaying infinite series.
- Algebraic techniques for simplification in symbolic identities.
- Pedagogical value of constructed mathematical relations in advanced analysis.
Excerpt from the Book
Methodology
The paper can be organized into three analytical steps.
Evaluation of the complex power
The expression (-1)^(-i/π) is interpreted through
a^b = e^(b log a)
where log denotes the principal logarithm. Since log(-1) = iπ, we obtain
(-1)^(-i/π) = e^( (-i/π)(iπ) ) = e
This is the key step in the complex-analysis component of the paper.
Study of the infinite series
Let
S = sum from k = 1 to ∞ of √5 / (πk)^(5k)
The general term is
a_k = √5 / (πk)^(5k)
Because the denominator grows like (πk)^(5k), the terms go to zero extremely fast. In fact, the series converges absolutely and very rapidly. This means that only the first few terms contribute meaningfully to numerical approximation.
Summary of Chapters
Introduction: Provides the definition of Euler's number and introduces the constructed identity under study, highlighting its components.
Methodology: Outlines the three analytical steps used to deconstruct the formula, including evaluation, series analysis, and algebraic simplification.
Discussion: Explains the pedagogical benefits of the identity and its value in demonstrating advanced mathematical themes like branch selection and symbolic cancellation.
Conclusion: Summarizes the findings, noting that the formula simplifies exactly to e and emphasizes the utility of such constructions for educational purposes.
Keywords
Euler's number, complex analysis, infinite series, transcendental constant, principal logarithm, branch selection, absolute convergence, algebraic simplification, symbolic identity, mathematical pedagogy, complex exponentiation, transcendental components.
Frequently Asked Questions
What is the primary focus of this research paper?
The paper focuses on the analysis and verification of a specific constructed identity that results in Euler's number (e).
What are the central themes discussed in the work?
The central themes include complex exponentiation, the behavior of rapidly convergent infinite series, and the importance of branch selection in complex analysis.
What is the core research objective?
The objective is to demonstrate how complex, seemingly non-trivial mathematical expressions can be simplified to fundamental constants through exact algebraic interpretation.
Which mathematical method is employed for the proof?
The author uses the principal branch of the complex logarithm to evaluate complex powers and demonstrates the algebraic cancellation of a redundant infinite series factor.
What topics are covered in the main section?
The main section covers the evaluation of the complex power, the study of the infinite series convergence, algebraic simplification of the full expression, and numerical verification.
Which keywords best characterize this study?
Key terms include Euler's number, complex analysis, absolute convergence, symbolic identity, and mathematical pedagogy.
How is the term (-1)^(-i/π) evaluated?
Using the identity a^b = e^(b log a) and the principal branch of the logarithm where log(-1) = iπ, the term evaluates precisely to e.
Why is the infinite series included if it cancels out?
The series is included to add visual complexity and to serve as an instructive example of how complicated-looking components can be structured to encode familiar constants.
- Citation du texte
- Fazal Rehman (Auteur), 2026, A Constructed Complex Identity Involving Euler’s Number and a Rapidly Convergent Infinite Series, Munich, GRIN Verlag, https://www.grin.com/document/1718801
