-
Title: Laurent Series
-
Series: Complex Analysis
-
YouTube-Title: Complex Analysis 15 | Laurent Series
-
Bright video: https://youtu.be/JH9eRAb1x7I
-
Dark video: https://youtu.be/lnkAj2tflt8
-
Ad-free video: Watch Vimeo video
-
Quiz: Test your knowledge
-
Dark-PDF: Download PDF version of the dark video
-
Print-PDF: Download printable PDF version
-
Thumbnail (bright): Download PNG
-
Thumbnail (dark): Download PNG
-
Subtitle on GitHub: ca15_sub_eng.srt missing
-
Timestamps
00:00 Intro
00:14 Generalization of power series
00:47 Radius of convergence
01:35 Consider inverse of complex numbers
02:20 Second radius of convergence for outside world
02:56 Series with negative powers is holomorphic
04:15 Two series combined gives holomorphic function
05:43 Definition Laurent Series
06:42 Principal part of Laurent Series
06:55 Residue of Laurent Series
07:20 Domain is a generalized ring
08:06 Credits
-
Subtitle in English (n/a)
-
Quiz Content
Q1: Let $\sum_{k=0}^\infty a_k z^k$ be a power series. Which value is not possible for the radius of convergence?
A1: $r = 0$
A2: $r = 0.5$
A3: $r = \pi$
A4: $r = \infty$
A5: $r = -\infty$
Q2: Let $\sum_{k=0}^\infty a_k z^k$ be a power series with radius of convergence $r = 2$. What is the domain of the holomomorphic function? $$w \mapsto \sum_{k=0}^\infty a_k \left( \frac{1}{w} \right)^k$$
A1: $\mathbb{C}$
A2: $\mathbb{C} \setminus \overline{B_{\frac{1}{2}}(0)}$
A3: $\mathbb{C} \setminus \overline{B_{2}(0)}$
A4: $\mathbb{C} \setminus B_{2}(0)$
Q3: Let $f(z) = \sum_{k=0}^\infty a_k z^k$ be a power series with radius of convergence $r = 2$. What is the derivative of the following function? $$g(w) = \sum_{k=0}^\infty a_k \left( \frac{1}{w} \right)^k$$
A1: $g^\prime(w) = f^\prime(w) $
A2: $g^\prime(w) = f^\prime(w^{-1}) $
A3: $g^\prime(w) = f^\prime(w^{-1}) \cdot \left( - \frac{1}{w^2} \right) $
A4: $g^\prime(w) = f^\prime(z) $
Q4: Let $f(z) = \sum_{k=0}^\infty a_k (z-z_0)^k$ be a power series with radius of convergence $r = 2$. What is the domain of the Laurent series $$\sum_{k=-2}^\infty a_k (z-z_0)^k $$ with $a_{-2} = a_{-1} = 1$?
A1: ${ z \in \mathbb{C} \mid 0 < | z - z_0 | < 2}$
A2: ${ z \in \mathbb{C} \mid \frac{1}{2} < | z - z_0 | < 2}$
A3: ${ z \in \mathbb{C} \mid \frac{1}{2} < | z - z_0 | < \infty}$
A4: ${ z \in \mathbb{C} \mid 0 < | z - z_0 | < \frac{1}{2}}$
-
Last update: 2024-10
