Intuition for i to the power i | Lockdown math ep. 9

From 3Blue1Brown.

i^i, visualized and explained.
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At 1:06:20, when changing r to equal 0.69*i, I said "this is what we might think of as (2i)^x", but that is not correct. It’s what we’d think of as [Exp(ln(2)*i)]^x for whatever complex number Exp(ln(2)*i) is.

Beautiful notes by Ngân Vũ

Video by Matt Parker:

Video by Red Pen Black Pen

Video timeline (thanks to user "nooonesperfect")

0:18 Exponential function for i^i
1:26 Question 1
2:27 Plug-in imaginary number in exp(x) polynomial
3:38 Answer 1 and explanation
7:35 What it really means i^i?
9:14 e^it as a position vector
11:30 Question 2
12:39 Audience question from twitter
13:14 Answer 2
14:52 Where you get after traveling π/2 units of time for position vector e^it
19:48 Question 3
20:42 Audience tweets
23:34 Answer 3
35:50 Question 4
37:11 Answer 4
40:11 How exp(rx) or b^x really works?
46:28 Question 5
47:49 Audience tweets
49:26 Answer 5
57:06 Visualization of f(x)= exp(r*x) i.e. e^(r*x), where r= unique complex number
1:06:06 Questions to think about
1:08:51 Audience tweets
1:09:09 Power tower for i
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Music by Vincent Rubinetti.
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