Complex Analysis Test - T 2

Rsquared Mathematics

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Time: 45:00

Topic: Complex Analysis

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The Quiz contains a total of 10 Questions.
Part B(Single Correct) contains 5 questions, each of 3 marks and -0.75 marks for incorrect answers.
Part C(MSQ) contains 5 questions, each with 4.75 marks and no negative marking. The maximum score for the Test is 38.75. Best of Luck.

Part - B.

1. What is the value of \(\sqrt{i} \times \sqrt{-i}\)

(a.) \(-\sqrt{2}\)
(b.) \(\sqrt{2}\)
(c.) \(i\)
(d.) \(-i\)

Explanation: \(=\exp\left(\frac{i \pi}{4}\right)+\exp\left(-\frac{i \pi}{4}\right)\)

2. Let \(f(z)=\frac{1}{z^2+1}\) and let \(p(z)=\sum_{n=0}^{\infty} a_n(z-a)^n\) be the power series expansion of \(f(z)\) about the point \(a \in \mathbb{R}\). Then the radius of convergence of \(P(z)\) is

(a.) \(\infty\)
(b.) \(|a|\)
(c.) \(\sqrt{a^2+1}\)
(d.) \(\sqrt{a^2-1}\)

Explanation: The radius of convergence of \(P(z)=\min \left\{|a-i|, |a+i|\right\}\)

3. Suppose \(f\) is analytic at \(z=z_0\) with \(f\left(z_0\right) \neq 0\) and \(g\) has a simple zero. \(\operatorname{Res}\left[\frac{f(z)}{g(z)} ; z_0\right]=\)

(a.) \(\frac{f\left(z_0\right)}{g\left(z_0\right)}\)
(b.) \(\frac{f^{\prime}\left(z_0\right)}{g\left(z_0\right)}\)
(c.) \(\frac{f\left(z_0\right)}{g^{\prime}\left(z_0\right)}\)
(d.) \(\frac{f^{\prime}\left(z_0\right)}{g^{\prime}\left(z_0\right)}\)

Explanation: \(\operatorname{Res}\left[\frac{f(z)}{g(z)} ; z_0\right]=\lim _{z \rightarrow z_0}\left(z-z_0\right) \frac{f(z)}{g(z)}\)

4. Consider \(f(z)=e^{\frac{-1}{z^2}} \cos \left(\frac{1}{z}\right)\) then find \(\operatorname{Res}[f(z): 0]=\)

(a.) 0
(b.) 1
(c.) -1
(d.) None of the above

Explanation: \(f\) is an even function, so the residue is 0.

5. Calculate the integral \(I=\int_{|z|-1}|z-1|dz\)

(a.) 8
(b.) 4
(c.) \(8 \pi\)
(d.) \(4 \pi\)

Explanation: Since \( |z| = 1 \), let \( dz = i e^{i \theta} d\theta \).
\(\Rightarrow |dz| = |d\theta| = d\theta \).
Now, \( I = \int_0^{2\pi} |\cos \theta + i \sin \theta - 1| d\theta \).
\(= \int_0^{2\pi} \sqrt{\cos^2 \theta + 1 - 2\cos \theta + \sin^2 \theta} d\theta\).

Part - C

6. Let \(D\) be an open connected subset of \(\mathbb{C}\). Let \(f\) be an analytic function on \(D\). Define \(S_1=\left\{z \in D: f(z)=0\ \text{or}\ f^{\prime}(z)=0\right\}\) then -

(a.) If \(S_1=D\) then \(f\) is constant
(b.) If \(S_1=\bar{D}\) then \(f\) is constant
(c.) If \(S_1 \neq D\) then \(f\) is non-constant
(d.) If \(S_1 \varsubsetneqq D\) then \(f\) can be constant

Explanation: By identity theorem, If \(S_1\) is uncountable, then \(f\) is constant. Also, f is contant iff \(S_1=D\)

7. Let \(f(z)=\left\{\begin{array}{c}(\frac{\bar{z}}{z})^2 ; z \neq 0 \\ 0, z=0\end{array}\right.\). Then which of the following are incorrect -

(a.) \(f\) is continuous at 0
(b.) \(f\) is analytic at 0
(c.) \(f\) is differentiable at 0
(d.) \(f\) satisfies \(C-R\) equation at 0

Explanation: \(f\) does not have a limit at 0.

8. Suppose that \(f(z)=u+i v\) be an entire function which satisfies the property \(u_y-v_x=-2 \ \forall z \in \mathbb{C}\). Then

(a.) \(f(z)\) is a constant function
(b.) \(f^{\prime}(z)\) is a constant function
(c.) \(f(z)\) is a polynomial function
(d.) \(f^{\prime}(z)\) is a polynomial function

Explanation: Use the \(C-R\) equation to claim that \(f^{\prime}\) is constant.

9. Consider \(P(z)=z^5-12 z^2+14\) and the region \(R_1=\{z \in \mathbf{C} \| |z|<1\}\) and \(R_2=\{z \in \mathbf{C} \| |z|>2\}\). Then

(a.) \(\mathrm{P}(\mathrm{z})\) has no root in \(R_1\)
(b.) \(\mathrm{P}(\mathrm{z})\) has 1 root in \(R_1\)
(c.) \(\mathrm{P}(\mathrm{z})\) has 2 roots in \(R_2\)
(d.) \(\mathrm{P}(\mathrm{z})\) has 3 roots in \(R_2\)

Explanation: Use Rouche's theorem.

10. Consider \( f(z) = \left(z^2 + 1\right)^{1/2} \) and let -
\( C_1 = \) simple closed curve enclosing \( i \) but not \( -i \)
\( C_2 = \) simple closed curve enclosing \( -i \) but not \( i \)
\( C_3 = \) Simple closed curve enclosing both \( i \) and \( -i \)
Let \( \Delta_{C_k} \arg f = \) change in \( \arg f \) when \( z \) goes around \( C_k \)

(a.) \(\Delta_{C_1} \arg f=\pi\)
(b.) \(\Delta_{C_2} \arg f=2 \pi\)
(c.) \(\Delta_{C_3} \arg f=2 \pi\)
(d.) \(\Delta_{C_3} \arg f=0 \)

Explanation: Change in \(\arg (f)=\) number of zeros inside the curve.