Sophomore's dream function: asymptotics, complex plane behavior and relation to the error function
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911426210168832 |
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| author | Irkhin, V. Yu. |
| author_facet | Irkhin, V. Yu. |
| contents | Sophomore's dream sum $S=\sum_{n=1}^\infty n^{-n}$ is extended to the function $f(t,a)=t\int_{0}^{1}(ax)^{-tx}dx$ 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 Im$t_n \simeq 2(n-1/8)πe/a$ for $a \geq 1$. The results obtained describe analytical integration of the function $x^{tx}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_10936 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sophomore's dream function: asymptotics, complex plane behavior and relation to the error function Irkhin, V. Yu. Classical Analysis and ODEs History and Overview Sophomore's dream sum $S=\sum_{n=1}^\infty n^{-n}$ is extended to the function $f(t,a)=t\int_{0}^{1}(ax)^{-tx}dx$ 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 Im$t_n \simeq 2(n-1/8)πe/a$ for $a \geq 1$. The results obtained describe analytical integration of the function $x^{tx}$. |
| title | Sophomore's dream function: asymptotics, complex plane behavior and relation to the error function |
| topic | Classical Analysis and ODEs History and Overview |
| url | https://arxiv.org/abs/2501.10936 |