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.
- Quote paper
- Fazal Rehman (Author), 2026, A Constructed Complex Identity Involving Euler’s Number and a Rapidly Convergent Infinite Series, Munich, GRIN Verlag, https://www.hausarbeiten.de/document/1718801
