Complex Analysis Test - T 3

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.) If \(f(z) \& \overline{f(z)}\) both are analytic on Domain \(\mathbb{D}\) then

(a.) There does not exist any closed set contained in \(f(\mathbb{D})\)
(b.) There exists a non-empty open set contained in \(f(\mathbb{D})\)
(c.) There exists an open set containing \(f(\mathbb{D})\)
(d.) None of these

Explanation:

\(f\) is constant.

(2.) \(f(z)=\left\{\begin{array}{l}z^2+1, z \neq 0 \\ 3, z=0\end{array}\right.\) then

\(\int_{|z|-1} f(z) dz = \ ?\)

(a.) 0
(b.) 1
(c.) \(2 \pi i\)
(d.) \(4 \pi i\)

Explanation:

\(f\) has a removable singularity, hence the integral is 0.

(3.) Let \(f: \mathbb{C} \rightarrow \mathbb{C}\) be a complex-valued function such that \(f^{(n)}(0)=3^n\) for \(n-\) even and \(f^{(n)}(0)=(n-1)!\) for \(n-\) odd, then -

(a.) \(f\) is entire
(b.) \(f\) is analytic in \(|z|<1\)
(c.) \(f\) is analytic in \(|z|>1\)
(d.) No such \(f\) exists

Explanation:

Evaluate the ROC of the series of \(f\).

(4.)

\[I=\frac{1}{2 \pi i} \int_{|z|-2} \frac{z}{(z-3)(z^n-1)} dz; \quad n \in \mathbb{N} \& n \geq 5\]

Then \(I = ?\)

(a.) \(\frac{9}{3^n-1}\)
(b.) \(\frac{9}{1-3^n}\)
(c.) \(\frac{18}{3^n-1}\)
(d.) \(\frac{18}{1-3^n}\)

Explanation:

Use the residue theorem.

(5.) Let \(f\) be a bilinear transformation such that \(f(z+1)=1+f(z) \forall z \neq 0 \& f(0)=0\) then

(a.) \(f\) maps the upper half plane onto a lower half plane
(b.) \(f\) maps the upper half plane onto the upper half plane
(c.) \(f\) maps the unit disk to the exterior of the unit disk
(d.) All of the above can be true

Explanation:

It has three fixed points, must be the identity map.

Part - C

(6.) For a complex-valued function \(f(z), z=z_0\) is a removable singularity if and only if -

(a.) \(f\) is bounded in a deleted neighborhood of \(z_0\)
(b.) \(f\left(z_0\right) \& \lim _{z \rightarrow z_0} f(z)\) both exist
(c.) Laurent series expansion around \(z_0\) contains only non-negative powers of \(z_0\)
(d.) None of the above

Explanation:

These all are one-way statements.

(7.) Let \(C\) be the circle \(|z|=1\) oriented in the counterclockwise direction. Then choose the correct integrals -

(a.) \(\int \frac{1}{z^2-8 z+1} dz=\frac{\pi i}{\sqrt{15}}\)
(b.) \(\int \frac{1}{x^2-8 z+1} dz=\frac{-\pi i}{\sqrt{15}}\)
(c.) \(\int_0^\pi \frac{1}{4-\cos \theta} d \theta=\frac{\pi}{\sqrt{15}}\)
(d.) \(\int_0^\pi \frac{1}{4-\cos \theta} d \theta=\frac{-\pi}{\sqrt{15}}\)

Explanation:

\(z^2-8z+1=(z-4-\sqrt{15})(z-4+\sqrt{15})\)

Hence, \(z^2-8z+1\) has a singularity inside \(C\) at \(z=4-\sqrt{15}\).

\(\int_C \frac{dz}{z^2-8z+1} = \frac{-\pi i}{\sqrt{15}}\)

Since \(|z|=1\), put \(z=e^{j\theta}\), then evaluate the last integral.
\(\Rightarrow \int_c \frac{d z}{z^2-8 z+1}=\int_{-\pi}^\pi \frac{d e^{j \theta}}{e^{20 \theta}-8 e^{j \theta}+1}\)
\(=\int_{-\pi}^\pi \frac{i e^{i \theta} d \theta}{e^{2 i \theta}-8 e^{i \theta}+1}\)
\(=i \int_{-\pi}^\pi \frac{d \theta}{e^{j \theta}+e^{-i \theta}-8}\)
\(=\frac{-i}{2} \int_{-\pi}^\pi \frac{d \theta}{4-\cos \theta}\)
\(=-i \int_0^\pi \frac{d \theta}{4-\cos \theta}\)

(8.) Let \(f(z)\) be defined as

\[ f(z)= \begin{cases}\frac{z^2+3 i z-2}{z+i}, & z \neq-i \\ 5, & z=-i\end{cases} \]

Then which of the following are correct -

(a.) \(f\) has a removable discontinuity at \(i\)
(b.) \(f\) is continuous at \(i\)
(c.) \(\lim _{z \rightarrow 0} f(z)\) does not exist
(d.) None of these

Explanation:

\(\lim _{z \rightarrow i} f(z) = \lim _{\substack{x \rightarrow 0 \\ y \rightarrow-1}} \frac{(x+i y)^2+3 i(x+i y)+2}{x+i y+1} = i \neq f(-i)\)

(9.) Which of the following is/are true -

(a.) There exists a continuous, periodic, and unbounded function on \(\mathbb{C}\)
(b.) There does not exist any function that is continuous, periodic, and unbounded on \(\mathbb{C}\)
(c.) Every function that is continuous and has first-order continuous partial derivatives in \(\mathbb{R}^2\) is harmonic
(d.) None of these

Explanation:

For (a) and (b), take \(\exp (z)\), and for (c), take \(u(x, y)=y^{\prime 2}-x\)

(10.) Let \(p(z)\) and \(q(z)\) be polynomials with no common zeroes, and with degrees \(m \& n\) respectively. Let \(f(z)=\frac{p(z)}{q(z)}\), then

(a.) \(f\) has a removable singularity at \(\infty\) if \(n \geq m\)
(b.) \(f\) has a removable singularity at \(\infty\) if \(m \geq n\)
(c.) \(\operatorname{Res}[f(z) ; \infty]=0\) if \(n>m+2\)
(d.) \(\operatorname{Res}[f(z) ; \infty]=0\) if \(m>n+2\)

Explanation:

Let \(g(z)=f\left(\frac{1}{z}\right)\).

\(g\) has \(m\) poles and \(n\) zeroes.

Hence, \(g(z)\) has a removable singularity at \(z=0\) if \(n \geq m+2\).

\(\operatorname{Res}[f(z) ; \infty]=\lim _{z \rightarrow 0} \frac{1}{z^2} f\left(\frac{1}{z}\right) = 0\) if \(n>m+2\).