Euler's Equation

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