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Title: Complex Logarithm
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Series: Complex Analysis
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YouTube-Title: Complex Analysis 13 | Complex Logarithm
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Original video for YT-Members (bright): Watch on YouTube
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Original video for YT-Members (dark): Watch on YouTube
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ca13_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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Timestamps
00:00 Intro
00:12 Logarithm as inverse of exponential function
00:33 How it works in the real numbers
01:54 Complex exponential function
03:08 exp(iy) lies on the unit circle
04:00 Definition of π
06:05 Exponential is 2πi-periodic
06:49 Exp is not injective
08:40 Restrict to open strip and slotted plane
09:06 Definition of logarithm (principal value)
09:56 Properties of logarithm
12:22 Credits
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Subtitle in English (n/a)
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Quiz Content
Q1: Which claim is not correct?
A1: $$ \exp \left( i \frac{\pi}{2} \right) = i$$
A2: $$ \exp \left( i \pi \right) = -1$$
A3: $$ \exp \left( i \frac{3 \pi}{2} \right) = -i$$
A4: $$ \exp \left( i 2 \pi \right) = -1$$
Q2: What is $\exp(i y)$ for $y \in \mathbb{R}$?
A1: $\cos(y) + \sin(y)$
A2: $\cos(y) + i \sin(y)$
A3: $\cos(y) - \sin(y)$
A4: $\cos(y) - i \sin(y)$
Q3: Let $\log$ be the principal value of the logarithm. What is $\log(i)$?
A1: $0$
A2: $\frac{\pi}{2}$
A3: $i \frac{\pi}{2}$
A4: $i \frac{3 \pi}{2}$
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Last update: 2024-10
