-
Title: Cauchy’s Integral Formula
-
Series: Complex Analysis
-
YouTube-Title: Complex Analysis 27 | Cauchy’s Integral Formula
-
Bright video: https://youtu.be/hll0DAilhoA
-
Dark video: https://youtu.be/54zwCAD5fds
-
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: ca27_sub_eng.srt missing
-
Timestamps (n/a)
-
Subtitle in English (n/a)
-
Quiz Content
Q1: Let $f: D \rightarrow \mathbb{C}$ be a holomorphic function where $\overline{B_r(z_0)} \subseteq D$. What is the correct Cauchy’s integral formula?
A1: $$f(z) = \frac{1}{2 \pi i} \oint_{\partial B_r(z_0) } \frac{ f(\zeta) }{ \zeta - z } d\zeta $$ for all $z \in B_r(z_0)$.
A2: $$f(z) = \frac{1}{2 \pi i} \oint_{\partial B_r(z_0) } \frac{ f(\zeta) }{ z - \zeta } d\zeta $$ for all $z \in B_r(z_0)$.
A3: $$f(z) = \frac{1}{2 \pi i} \oint_{\partial B_r(z_0) } \frac{ f^\prime(\zeta) }{ \zeta - z } d\zeta $$ for all $z \in B_r(z_0)$.
A4: $$f(z) = \frac{1}{2 \pi i} \oint_{\partial B_r(z_0) } \frac{ f^\prime(\zeta) }{ \zeta - z } d \xi $$ for all $\zeta \in B_r(z_0)$.
Q2: Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic function given by $f(z) = z^2$. What is correct by applying Cauchy’s integral formula?
A1: $$f(z) = \frac{1}{2 \pi i} \oint_{\partial B_1(0) } \frac{ \zeta^2 }{ \zeta - z } d\zeta $$ for all $z \in B_1(0)$.
A2: $$f(z) = \frac{1}{2 \pi i} \oint_{\partial B_1(0) } \frac{ \zeta}{ \zeta - z } d\zeta $$ for all $z \in B_1(0)$.
A3: $$f(z) = \frac{1}{2 \pi i} \oint_{\partial B_1(0) } \frac{ 1 }{ \zeta - z } d\zeta $$ for all $z \in B_1(0)$.
A4: $$f(z) = \frac{1}{2 \pi i} \oint_{\partial B_1(0) } \frac{ \zeta }{ \zeta^2 - z } d\zeta $$ for all $z \in B_1(0)$.
-
Last update: 2024-10
