Uploaded February 2023 | Updated September 2026, 2 days ago
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 - amzn.to/3MDHUis
Gerald Folland - Real Analysis: Modern Techniques and Their Applications - amzn.to/3JmK6Np
R. Shankar - Principles of Quantum Mechanics - amzn.to/40825xj
Maxwell Rosenlicht - Introduction to Analysis - amzn.to/3NFYiAC
Tom Apostol - Mathematical Analysis 2nd Edition - 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 youtu.be/v5rD0B-zfXw
Mastering the Art of Reading a Proof: By Example youtu.be/WPWf_s6Q1QQ
Introduction to Control Theory youtu.be/0v4WFmOm764
//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:
epidemicsound.com/referral/644nao
//Recording Equipment
Canon SL3: amzn.to/3nZ11KU
Canon T6i: amzn.to/3FUpkQh
Rode VideoMic: amzn.to/3lhldGa
Blue Yeti Microphone: amzn.to/3I1y88N
Yeti Nano Microphone: amzn.to/3I1mriA
SanDisc 256GB SD Card: amzn.to/3E3LOOr
Neewer 5600K USB LED Lights: amzn.to/3xvB9cN
Neewer 18 inch Ring Light: amzn.to/2ZvgCsc
Camera Power Adapter: 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
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 - amzn.to/3MDHUis
Gerald Folland - Real Analysis: Modern Techniques and Their Applications - amzn.to/3JmK6Np
R. Shankar - Principles of Quantum Mechanics - amzn.to/40825xj
Maxwell Rosenlicht - Introduction to Analysis - amzn.to/3NFYiAC
Tom Apostol - Mathematical Analysis 2nd Edition - 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 youtu.be/v5rD0B-zfXw
Mastering the Art of Reading a Proof: By Example youtu.be/WPWf_s6Q1QQ
Introduction to Control Theory youtu.be/0v4WFmOm764
//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:
epidemicsound.com/referral/644nao
//Recording Equipment
Canon SL3: amzn.to/3nZ11KU
Canon T6i: amzn.to/3FUpkQh
Rode VideoMic: amzn.to/3lhldGa
Blue Yeti Microphone: amzn.to/3I1y88N
Yeti Nano Microphone: amzn.to/3I1mriA
SanDisc 256GB SD Card: amzn.to/3E3LOOr
Neewer 5600K USB LED Lights: amzn.to/3xvB9cN
Neewer 18 inch Ring Light: amzn.to/2ZvgCsc
Camera Power Adapter: 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
