Estimates of the minimum of the Gamma function using the Lagrange inversion theorem and the Fa\`a di Bruno formula
Autor: | Pain, Jean-Christophe |
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Rok vydání: | 2024 |
Předmět: | |
Druh dokumentu: | Working Paper |
Popis: | In this article we derive, using the Lagrange inversion theorem and applying twice the Fa\`a di Bruno formula, an expression of the minimum of the Gamma function $\Gamma$ as an expansion in powers of the Euler-Mascheroni constant $\gamma$. The result can be expressed in terms of values the Riemann zeta function $\zeta$ of integer arguments, since the multiple derivative of the digamma function $\psi$ evaluated in $1$ is precisely proportional to the zeta function. The first terms (up to $\gamma^6$) were provided in order to address the convergence of the series. Applying the Lagrange inversion theorem at the value $3/2$ yields more accurate results, although less elegant formulas, in particular because the digamma function evaluated in $3/2$ does not simplify. |
Databáze: | arXiv |
Externí odkaz: |
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