Ecosystem Interactions Class Discussion Presentation in Blue Green Lined Styl...
Lesson 3bi proving euler’s equation without using power series
1. Proving Euler’s Equation without Using Power Series
Let us start with the following function: 𝑓 𝜃 = 𝑒!!"
(𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃)
and use calculus to prove Euler’s equation.
Find 𝑓′ 𝜃 by using the product rule.
What does the derivative of 𝑓′ 𝜃 equal to?
What does this imply about 𝑓 𝜃 ?
This means 𝑓 𝜃 = 𝑘
What is k when we substitute 𝜃 = 0?
This means 𝑓 𝜃 = 1
Now multiply both sides by 𝑒!"
Euler’s Form for a Complex Number