Solve your IGNOU Doubts

Question:

Evaluate the following integrals

a) $\int_{0}^{\infty} \frac{x^2 + 2}{(x^2 + 1)(x^2 + 4)} \, dx$ .
b) $\int_{-\infty}^{\infty} \frac{\sin^2 2x}{1 + x^2} \, dx$ .

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Question:

Show that there is only one linear fractional transformation that maps three given distinct points z₁, z₂ and z₃ in the extended z plane onto three specified distinct points w₁, w₂ and w₃ in the extended w plane.

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Question:

Find the image of the semi-infinite strip x > 0, 0 < y < 1 when $w = i/z$ . Sketch the strip and its image.

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Question:

Prove that the linear fractional transformation $\phi(z) = \frac{2z - 1}{2 - z}$ maps the circle $c : |z| = 1$ into itself. Also prove

that f(z) is conformal in $\bar{D} = \{z : |z| \leq 1\}$ .

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Question:

Find the zeros and singularities of the function $f(z) = \frac{z}{4 \cos^2 z - 1}$ in $|z| \leq 1$ . Also find the residue at the poles

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Question:

Expand $f(z) = \frac{1}{(z-1)^2 (z-3)}$ in a Laurent series valid for

i) 0 < |z - 1| < 2 and $\quad \quad \quad \quad$ ii) 0 < |z - 3| < 2.

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Question:

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

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Question:

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

Image ignouassignments-ignouacademy-com--p-ignou-12159

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Question:

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

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Question:

Evaluate $\oint_c \frac{dz}{z^2 + 1}$ where c is the circle $|z| = 4$ .

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Question:

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

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Question:

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

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Question:

If $p(z) = a_0 + a_1 z + \dots + a_{n-1} z^{n-1} + z^n (n \geq 1)$ , 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 $\sin z = 5$ .

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Question:

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 $r = 1$ ?

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Question:

) Evaluate the following integrals:
i) $I = \int_{0}^{2\pi} f(e^{i\theta}) \cos^2(\theta/2) \, d\theta$ .

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

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Question:

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

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Question:

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.

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Question:

If $f = u + iv$ is entire such that $u_x + v_y = 0$ in $\mathbb{C}$ then show that f has the form $f(z) = az + b$ where $a, \mathbf{b}$ are constants with $\operatorname{Re} a = 0$ .

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Question:

Determine whether each of the following statement is true or false. Justify your answer with a short proof or a counter example.
i) If $z = a + ib$ , where a and b are integers, then $|1 + z + z^2 + \dots + z^n| \geq |z|^n$ if a > 0.
ii) If f(z) and $\overline{f(\bar{z})}$ 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) $\lim_{n \to \infty} (n!)^{1/n} = \infty$ .
v) The inequality $|e^a - e^b| \leq |a - b|$ holds for $a, b \in D = \{w : \text{Re } w \leq 0\}$ .
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 |b_n| < n² |a_n| 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 $f(z) = g(z^2)$ for all $z \in \mathbb{C}$ .
ix) A mobius transformation which maps the upper half plane $\{z : \text{Im } z > 0\}$ onto itself and fixing $0, \infty$ and no other points, must be of the form $Tz = \alpha z$ for some $\alpha > 0$ and $\alpha \neq 1$ .
x) If f is entire and $\text{Re } f(z)$ is bounded as $|z| \to \infty$ , then f is constant.