MMT-005 Solved Assignment 2023

IGNOU MMT-005 Solved Assignment 2023 | M.Sc. MACS

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

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IGNOU MMT-005 Assignment Question Paper 2023

Course Code: MMT-005

Assignment Code: MMT-005/TMA/2023

 

1. Determine whether each of the following statement is true or false. Justify your answer with a short proof or a counter example.

i) If \(z=a+i b\), where \(a\) and \(b\) are integers, then \(\left|1+z+z^{2}+\cdots+z^{n}\right| \geq|z|^{n}\) if \(a>0\).

ii) If \(f(z)\) and \(\overline{f(z)}\) are analytic functions in \(a\) domain, then \(f\) is necessarily a constant.

iii) A real-valued function \(u(x, y)\) is harmonic in \(D\) iff \(u(x,-y)\) is harmonic in \(D\).

iv) \(\lim _{n \rightarrow \infty}(n !)^{1 / n}=\infty\)

v) The inequality \(\left|e^{a}-e^{b}\right| \leq|a-b|\) holds for \(a, b \in D=\{w\) : Re \(w \leq 0\}\).

vi) If \(f(z)=\sum_{n=0}^{\infty} a_{n}(z-a)^{n}\) has the property that \(\sum_{n=0}^{\infty} f^{(n)}(a)\) converges, then \(f\) is necessarily an entire function.

vii) If a power series \(\sum_{n=0}^{\infty} a_{n} z^{n}\) converges for \(|z|<1\) and if \(b_{n} \in \mathbb{C}\) is such that \(\left|b_{n}\right|<n^{2}\left|a_{n}\right|\) for all \(n \geq 0\), then \(\sum_{n=0}^{\infty} b_{n} z^{n}\) converges for \(|z|<1\).

viii) If \(f\) is entire and \(f(z)=f(-z)\) for all \(z\), then there exists an entire function g such that \(f(z)=g\left(z^{2}\right)\) for all \(z \in \mathbb{C}\).

ix) A mobius transformation which maps the upper half plane \(\{z: \operatorname{Im} z>0\}\) onto itself and fixing \(0, \infty\) and no other points, must be of the form \(T z=\alpha z\) for some \(\alpha>0\) and \(\alpha \neq 1\).

x) If \(f\) is entire and \(\operatorname{Re} f(z)\) is bounded as \(|z| \rightarrow \infty\), then \(f\) is constant.

2. a) If \(f=u+i v\) is entire such that \(u_{x}+v_{y}=0\) in \(\mathbb{C}\) then show that \(f\) has the form \(f(z)=a z+b\) where \(a, \mathbf{b}\) are constants with \(\operatorname{Re} a=0\).

b) Consider \(f(z)=z^{2}-z\) and the closed circular region \(R=\{z:|z| \leq 1\}\). Find points in \(R\) where \(|f(z)|\) has its maximum and minimum values.

c) Find the points where the function \(f(z)=\frac{\log (z+4)}{z^{2}+i}\) is not analytic.

3. a) Evaluate the following integrals:

i) \(I=\int_{0}^{2 \pi} f\left(e^{i \theta}\right) \cos ^{2}(\theta / 2) d \theta\).
ii) \(I=\int_{0}^{2 \pi} f\left(e^{i \theta}\right) \sin ^{2} \theta / 2 d \theta\)

b) Find the image of the circle \(|z|=r(r \neq 1)\) under the mapping \(w=f(z)=\frac{z-i}{z+i}\). What happens when \(r=1\) ?

4. a) If \(p(z)=a_{0}+a_{1} z+\cdots+a_{n-1} z^{n-1}+z^{n}(n \geq 1)\), then show that there exists a real \(R>0\) such that \(2^{-1}|z|^{n} \leq|p(z)| \leq 2|z|^{n}\) for \(|z| \geq R\).

b) Find all solutions to the equation \(\sin z=5\).

5. a) Find the constant \(c\) such that \(f(z)=\frac{1}{z^{n}+z^{n-1}+\cdots+z^{2}+z^{-n}}+\frac{c}{z-1}\) can be extended to be analytic at \(z=1\), when \(n \in \mathbb{N}\) is fixed.

b) Find all the singularities of the function \(f(z)=\exp \left(\frac{z}{\sin z}\right)\).

c) Evaluate \(\oint_{C} \frac{d z}{z^{2}+1}\) where \(c\) is the circle \(|z|=4\).

6. a) Find the maximum modulus of \(f(z)=2 z+5 i\) on the closed circular region defined by \(|z| \leq 2\).

b) Evaluate \(\int_{C} \frac{z^{3}+3}{z(z-i)^{2}} d z\), where \(c\) is the eight like figure shown in Fig. 1.

![](https://cdn.mathpix.com/cropped/2023_04_04_7e6f3a22a614e7cab45bg-4.jpg?height=668&width=983&top_left_y=1571&top_left_x=545)

Fig. 1

c) Find the radius of convergence of the following series.
i) \(\quad \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k !}(z-1-i)^{k}\)
ii) \(\quad \sum_{k=1}^{\infty}\left(\frac{6 k+1}{2 k+5}\right)^{k}(z-2 i)^{k}\)

\(2\:cos\:\theta \:cos\:\phi =cos\:\left(\theta +\phi \right)+cos\:\left(\theta -\phi \right)\)

MMT-005 Sample Solution 2023

 

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\(sin\left(\theta -\phi \right)=sin\:\theta \:cos\:\phi -cos\:\theta \:sin\:\phi \)

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