Ivan Shishkin, Rye (1878)

Problems/Real analysisUnreviewed

Résoudre xx=y\displaystyle x^{x}=y pour e−e−1⩽y⩽ee−1\mathrm{e}^{-\mathrm{e}^{-1}}\leqslant y\leqslant\mathrm{e}^{\mathrm{e}^{-1}}

by mathman·
70
Difficulty scaleÉchelle de difficulté

This score reflects both the level of the required concepts and the difficulty of the solution.Ce score tient compte à la fois du niveau des notions nécessaires et de la difficulté de la résolution.

  1. 1–10First steps / middle schoolPremiers pas / collège
  2. 11–25Beginner / high schoolDébutant / lycée
  3. 26–50Intermediate / undergraduateIntermédiaire / licence
  4. 51–70Advanced / graduateAvancé / master
  5. 71–90Expert / specializedExpert / spécialisé
  6. 91–100Research levelNiveau recherche
These levels are approximate guides.Ces niveaux sont des repères approximatifs.
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Pour yy réel, tel que e−e−1⩽y⩽ee−1\mathrm{e}^{-\mathrm{e}^{-1}}\leqslant y\leqslant\mathrm{e}^{\mathrm{e}^{-1}}

Montrez que :

x=1+ln⁡(y)+∑k=1+∞(−1)kkk(k+1)!ln⁡(y)k+1\displaystyle x=1+\ln(y)+\sum_{k=1}^{+\infty}(-1)^{k}\frac{k^{k}}{(k+1)!}\ln(y)^{k+1}

est solution de xx=y\displaystyle x^{x}=y

(et ce n’est pas le développement en série de la fonction de Lambert).

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