[EM#34] Holomorphie des fonctions analytiques (Démonstration)

In this episode, I essentially demonstrate that any function expandable as a power series is holomorphic on its disk of convergence, or simply "differentiable," if we consider only a function of the real variable. This proof is a great opportunity to showcase several modes of convergence for series of functions, and Marcel didn't miss it… ✍🏻 Erratum: 15:13 – Typo in the recap: the variable is h, not z! 🕒 Time markers: 0:00 – Presentation of the theorem 1:30 – Common step – Notable identity 4:27 – Conjecture and contextualization 7:20 – Proof #1 – Back to basics! 14:03 – Proof #2 - Uniform Convergence 17:02 – About the Two Proofs 🎥 Related Episodes: [GS#3]    • [GS#3] Quatre modes de convergence des sér...   [UT#71]    • [UT#71] Les séries entières (Introduction)   [EM#1]    • [EM#1] Une fameuse identité remarquable (D...   [CM#6]    • [CM#6] Lettre d'Abel à propos de la conver...   [ETI#4]    • [ETI#4] Limite ponctuelle d'une suite de f...   [EM#21]    • [EM#21] Limite uniforme d'une suite de fon...   ✒️ Concepts Covered: power series, radius of convergence, open ball, holomorphic function, analytic function, notable identity, rate of change, term-by-term differentiation, remainder of a convergent series, convergence of series of functions, convergence Uniform, continuity, normal convergence. 🌞 Enjoy listening! 🌐 ​​Explore my website! – https://www.oljen.fr/ 📚 Discover my training courses! – https://www.oljen.fr/formations/ 🤝🏻 You can make a donation here! – https://www.paypal.com/donate/?hosted... In this [EM] series, I revisit essential mathematical proofs from higher education. Beyond the proofs themselves, I try to explain the "why" and "how": where do the essential ideas come from? What is the purpose of the proof? Which key ideas are worth remembering? #BacPlus2 #Analysis #Proof