Defining Exponentials the "EASY" and HARD way скачать в хорошем качестве

Defining Exponentials the "EASY" and HARD way

The exponential function is fundamental to all of mathematics, but before you can use it, you have to define it first. Here we build the exponential function in two ways. One way uses the least upper bound property, and another leverages Picard's Theorem and Initial Value Problems. //Books Walter Rudin - Principles of Mathematical Analysis 3rd Edition - https://amzn.to/3MDHUis Gerald Folland - Real Analysis: Modern Techniques and Their Applications - https://amzn.to/3JmK6Np R. Shankar - Principles of Quantum Mechanics - https://amzn.to/40825xj Maxwell Rosenlicht - Introduction to Analysis - https://amzn.to/3NFYiAC Tom Apostol - Mathematical Analysis 2nd Edition - https://amzn.to/3mwxFlC //Exercises Prove the product to sum identities for exponentials first for rational numbers then for real numbers. //Watch Next The Real Analysis Survival Guide    • The Real Analysis Survival Guide   Mastering the Art of Reading a Proof: By Example    • Mastering the Art of Reading Proofs: By Ex...   Introduction to Control Theory    • Introduction to Control Theory   //Music Provided by Epidemic Sound Between the Lines - Elijah N Coffeehouse - Large Beach Memories - Sum Wave bam. bam. - Yomoti Step it up - Riverworm Future Vibes - Qeeo Quixotic - Jobii Use this referral link to get a 30 day free trial with Epidemic Sound for your YouTube channel: https://www.epidemicsound.com/referra... //Recording Equipment Canon SL3: https://amzn.to/3nZ11KU Canon T6i: https://amzn.to/3FUpkQh Rode VideoMic: https://amzn.to/3lhldGa Blue Yeti Microphone: https://amzn.to/3I1y88N Yeti Nano Microphone: https://amzn.to/3I1mriA SanDisc 256GB SD Card: https://amzn.to/3E3LOOr Neewer 5600K USB LED Lights: https://amzn.to/3xvB9cN Neewer 18 inch Ring Light: https://amzn.to/2ZvgCsc Camera Power Adapter: https://amzn.to/3D3upUu DISCLAIMER: The links above in this description may be affiliate links. If you make a purchase with the links provided I may receive a small commission, but with no additional charge to you :) Thank you for supporting my channel so that I can continue to produce mathematics content for you! 0:00 Introduction 1:32 Chill Vibes 2:05 Rational Exponents 5:37 Least Upper Bound for Rationals 10:47 Real Exponents 11:44 The Easy Way ODEs 14:17 Wrap Up