• 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

• Quiz: Test your knowledge

• 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 $z \in B_1(0)$.

• Back to overview page