Quiz 33 Description of an arc Let R0 Which of the following five statements gives a correct description of the arc with parametrisation t R en t for t 2 2 A is a semi circular arc centred at 0 in the...

### Solution By Steps #### Quiz 3.3. ***Step 1: Understand Parametrization*** The parametrization $\Gamma(t) = Re^{nt}$ for $t \in [-\pi/2, \pi/2]$ is incorrect as stated for describing a semi-circular arc. The correct form for a semi-circular arc should involve $e^{it}$, not $e^{nt}$. Assuming the correct form is $\Gamma(t) = Re^{it}$, we proceed. ***Step 2: Analyze Correct Form*** With $\Gamma(t) = Re^{it}$ for $t \in [-\pi/2, \pi/2]$, this describes a semi-circle in the complex plane, starting from $-iR$ to $iR$ through the right half-plane. #### Final Answer (D) $\Gamma$ is a semi-circular arc centred at 0 in the right half-plane. #### Key Concept Parametrization #### Key Concept Explanation Parametrization converts a complex number's representation from Cartesian (real and imaginary parts) to polar form (magnitude and angle), facilitating the description of curves in the complex plane. --- ### Solution By Steps #### Quiz 3.4. ***Step 1: Parametrize $\Gamma$*** Given $\Gamma(t) = Re^{it}$ for $t \in [-\pi/2, \pi/2]$, we have $z = Re^{it}$ and $dz = iRe^{it}dt$. ***Step 2: Substitute $z$ and $dz$*** Substitute $z$ and $dz$ into the integral $J = \int_{\Gamma} \frac{\log z}{z} dz$ to get $J = \int_{-\pi/2}^{\pi/2} \frac{\log(Re^{it})}{Re^{it}} iRe^{it} dt$. ***Step 3: Simplify Integral*** Simplify to $J = i\int_{-\pi/2}^{\pi/2} \log(Re^{it}) dt = i\int_{-\pi/2}^{\pi/2} (\log R + it) dt$. ***Step 4: Evaluate Integral*** Evaluate the integral: $J = i[\log R t + \frac{1}{2}it^2]_{-\pi/2}^{\pi/2} = i\pi\log R + i^2\frac{\pi^2}{2}$. ***Step 5: Simplify Final Answer*** Simplify to get $J = i\pi\log R - \frac{\pi^2}{2}$. #### Final Answer (E) $J = \frac{\pi^2}{2} + i\pi\log R$ #### Key Concept Complex Integration #### Key Concept Explanation Complex integration involves integrating functions over curves in the complex plane. Parametrization is used to express the curve in terms of a real parameter, simplifying the integration process.