Ivan Shishkin, Rye (1878)

Problems/Real analysisUnreviewed

Résoudre ab=ba\displaystyle a^{b}=b^{a}

by mathman·
50
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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Prenons 1<a⩽e⩽b\displaystyle 1<a\leqslant\mathrm{e}\leqslant b,

Montrez que les solutions de ab=ba\displaystyle a^{b}=b^{a} sont les couples (a;b)(a;b) tels que :

ln⁡(a)=∑k=1+∞kk−1k!(ln⁡(b)b)k\displaystyle\ln(a)=\sum_{k=1}^{+\infty}\frac{k^{k-1}}{k!}\left(\frac{\ln(b)}{b}\right)^{k}

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