2025/01/19 by Irkhin, V. Yu.
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #History and Overview (math.HO)
paper · doi:10.48550/arxiv.2501.10936
Sophomore's dream sum S=∑n=1^∞ n-n is extended to the function f(t,a)=t∫01(ax)-txdx with f(1,1)=S. Asymptotic behavior for a large |t| is obtained, which is exponential for t>0 and t<0,a>1, and inverse-logarithmic for t<0,a<1. An advanced approximation includes a half-derivative of the exponent and is expressed in terms of the error function. This approach provides excellent interpolation description in the complex plane. The function f(t,a) demonstrates for a>1 oscillating behavior along the imaginary axis with slowly increasing amplitude and the period of 2πiea, modulation by high-frequency oscillations being present. Also, f(t,a) has non-trivial zeros in the left complex half-plane with Imtn ≃ 2(n-1/8)πe/a for a>1. The results obtained describe analytical integration of the function xtx.