Contour Integral & Theorem MCQ Quiz in తెలుగు - Objective Question with Answer for Contour Integral & Theorem - ముఫ్త్ [PDF] డౌన్‌లోడ్ కరెన్

Explanation:

Since $\frac{3z^2+7z+1}{z+1}$ has singularity at z= -1 which is outside $|\mathrm{z}|=\frac{1}{2}$

Hence this function is analytic inside $|\mathrm{z}|=\frac{1}{2}$ and ${\int }_{\gamma }fd\gamma =0$

for analytic function f inside $\gamma$

Hence ${\int }_{\mathrm{c}}\frac{3{\mathrm{z}}^2+7\mathrm{z}+1}{\mathrm{z}+1}\mathrm{d}\mathrm{z}$ = 0

Therefore, option 1 is correct.

​Explanation:

On the unit circle C, the complex conjugate -z is related to z as

$3-\bar{z}=3-\frac{1}{z}$  

${\oint }_{\mathrm{c}}\frac{\mathrm{d}\mathrm{z}}{3-\bar{\mathrm{z}}}={\oint }_{\mathrm{c}}\frac{\mathrm{z}\mathrm{d}\mathrm{z}}{3\mathrm{z}-1}={\oint }_{\mathrm{c}}\frac{1}{3}.\frac{\mathrm{z}\mathrm{d}\mathrm{z}}{\mathrm{z}-1/3}$  

since ${\oint }_{\mathrm{c}}\frac{\mathrm{f}(\mathrm{z})\mathrm{d}\mathrm{z}}{\mathrm{z}-\mathrm{a}}=2\pi \mathrm{i}\mathrm{f}(\mathrm{a})$  

${\oint }_{\mathrm{c}}\frac{\mathrm{d}\mathrm{z}}{3-\bar{\mathrm{z}}}$ =  $\frac{1}{3}.2\pi i(\frac{1}{3})=\frac{2\pi i}{9}$  

Therefore, option 2 is correct.

Explanation:

 ${\int }_0^{\pi }\frac{1}{z}dz$  = ∫f(e^iθ) $\frac{d}{d\theta }$ (eiθ) dz

⇒ ${\int }_0^{\pi }$ $\frac{1}{e^{(}i\theta )}$i e^iθ dz  ⇒ πi

Hence, The Correct Answer is Option 3.