Complex Analysis Test - T 1

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. Which of the following is true about a meromorphic function -

(a.) Composition of meromorphic function is meromorphic
(b.) A meromorphic function can have infinite poles in any Domain
(c.) A meromorphic function can have an essential singularity in the extended complex plan
(d.) None of the above

Exp: For b, take any bounded domain, For c, \(f(z) = e^z\).

2. If \(a > e\), the equation \(e^z = a z^2\) has \(n\) real roots inside the unit circle. Then the value of \(n\) is

(a.) 1
(b.) 0
(c.) 2
(d.) None of the above

Exp: Use Rouche's theorem to claim.

3. The Radius of convergence of the power series \(\sum \log n \cdot(z)^{\log n}\) is

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

Exp: Limit of \((\log n)^{(1 / \log n)} = 1\)

4. \(\int_{|z|=10} \frac{3 z^{3}-z}{z^{3}-z+1} d z =\)

(a.) 0
(b.) \(2 \pi i\)
(c.) \(-2 \pi i\)
(d.) None of the above

Exp: If \(f\) is analytic, then \(\int z \frac{f^{\prime}(z)}{f(z)} d z = 2 \pi i\) (sum of root of roots of \(f\)).

5. Select Correct -

(a.) Under analytic function image of an open set may not be open.
(b.) An Entire function may map a bounded set to an unbounded set.
(c.) Analytic function maps a closed set to a closed set.
(d.) Limit point of the set of zeros of an analytic function is always a non-isolated essential singularity.

Exp: For (a), take a constant function. For (b), analytic function maps a bounded set to a bounded set. For (c), take the exponential function. For (d), \(\sin (1 / z)\) in the unit circle.

Part - C

6. Let \(f\) be an entire function such that \(f(\langle a_{n} \rangle) = \langle b_{n} \rangle\), where \(\langle a_{n} \rangle\) and \(\langle b_{n} \rangle\) are sequences of complex numbers, choose the corrects -

(a.) If \(\langle a_{n} \rangle\) is constant and \(\langle b_{n} \rangle\) is Bounded then \(f\) is constant
(b.) If \(\langle a_{n} \rangle\) is non-constant bounded sequence and \(\langle b_{n} \rangle\) is constant then \(f\) is constant
(c.) If Range \(\langle a_{n} \rangle\) has a limit point in \(\mathbb{C}\) and \(\langle b_{n} \rangle\) is constant then \(f\) is constant
(d.) If \(\langle a_{n} \rangle \rightarrow 0\) then \(f\) is constant.

Exp: Use Identity theorem for entire functions

7. Which of the following are correct -

(a.) For a function \(f\), if the residue at a point \(z_{0}\) is 0 then \(f\) has a removal singularity at point \(z_{0}\)
(b.) If \(f\) has a removal singularity at any point in \(\mathbb{C}\) then the residue at that point is 0.
(c.) If \(f\) has a pole of order 5 at a point, then the residue at that point can be zero
(d.) If \(f\) has a pole of order 5 at a point, then the residue at that point must be non-zero.

Exp: For a, c, d take \(f(z) = \frac{1}{z^{5}}, \ b\) is a true statement

8. Which of the following is true about an entire function -

(a.) Set of all entire functions forms an Integral domain with respect to standard addition and multiplication of functions
(b.) Set of all entire functions forms a Field with respect to standard addition and multiplication of functions
(c.) Set of all analytic functions forms an integral on a set \(D\) if and only if \(D\) is connected
(d.) If \(|f(z) - z_{0}| > c > 0\) for some \(z_{0} \in \mathbb{C}\) then \(f\) is constant.

Exp: For b, non-existence of the inverse.

9. Let \(f(z) = \frac{1}{1 - z} e^{-\frac{1}{z}}\), then choose the correct -

(a.) Residue at zero is \(\frac{1}{e}\)
(b.) Residue at zero is \(\frac{1 - e}{e}\)
(c.) There exist a sequence \(\langle a_{n} \rangle \in (0, 1)\) such that \(f(a_{n}) \rightarrow 2020\).
(d.) There exist a sequence \(\langle a_{n} \rangle \in (0, 1)\) such that \(f(a_{n})\) is divergent.

Exp: Find Laurent's series and evaluate the coefficient of \(1/z\), for c and d, use the Casorati-Weierstrass theorem.

10. Let \(f(z) = \left(\sinh \frac{1}{z}\right)^{-1}\) then

(a.) \(f\) has an infinite number of singularities
(b.) All singularities of \(f\) lie on the real axis
(c.) 0 is a non-isolated singularity of \(f\)
(d.) All the singularities of \(f\) are poles.

Exp: Set of singularities of \(f = \{0\} \cup \{1/n \pi i \mid n \in \mathbb{Z}-0\}\)