Joseph Heyward, Meaza Bogale, Daniel Ntiamoah and Alemayehu Negash

We study exact representations of truncated geometric sums raised to complex powers and derive closed-form
expressions for the associated complex exponents. Given polynomially generated complex data, we show that
the exponent 𝑠𝑚 satisfying 𝑍𝑚 = 𝑆𝑚(𝑥) 𝑠𝑚 can be recovered explicitly via logarithmic and trigonometric relations.
In this setting, the term derived complex exponent refers to the exponent obtained from the finite identity
𝑍𝑚 = 𝑆𝑚(𝑥) 𝑠𝑚 , rather than being specified a priori. When the data grow at most polynomially and 𝑥 > 1, the real
part of the derived exponent converges to zero at a logarithmic rate, while the imaginary part remains bounded
and oscillatory. The results are algebraic and asymptotic in nature and do not rely on analytic continuation,
Dirichlet series, or special-function theory. Numerical experiments illustrate exact recovery and the asymptotic
behavior of the derived exponents.