Complex Variables and Partial Differential Equations (3130005)

CVPDE-3130005

Complex variable functions

BE | Semester 3 Unit : Complex variable functions

Q1 (a) Winter-2019 Find the real and imaginary parts of $f (z) = \frac{3 i}{2 + 3 i}$ .

Unit : Complex variable functions

Q1 (a) Summer-2020 Find the analytic function $f (z) = u + iv, if u = x^{3} - 3 {xy}^{2}$

Unit : Complex variable functions

Q1 (b) Summer-2020 Find the roots of the equation, $z^{2} - (5 + i) z + (8 + i) = 0$ .

Unit : Complex variable functions

Q1 (b) Winter-2019 State De-Movire’s formula and hence evaluate ${(1 + i \sqrt{3})}^{100} + {(1 - i \sqrt{3})}^{100} .$

Unit : Complex variable functions

Q1 (c) Winter-2019 Define harmonic function. Show that $u (x, y) = \sinh x \sin y$ is a harmonic function. Find its harmonic conjugate $v (x, y)$ .

Unit : Complex variable functions

Q1 (c ( i )) Summer-2020 Determine & sketch the image $| z | = 1$ of under the transformation $w = z + i$ .

Unit : Complex variable functions

Q1 (c ( ii )) Summer-2020 Find the real and imaginary parts of $f (z) = z^{2} + 3 z$ .

Unit : Complex variable functions

Q2 (a) Winter-2019
Determine the Mobius transformation which maps $z_{1} = 0, z_{2} = 1, z_{3} = \infty$ into $, w_{1} = - 1, w_{2} = - i, w_{3} = 1$ .

Unit : Complex variable functions

Q2 (b) Winter-2019 Define $\log z$ , prove that $i^{i} = e^{- (4 n + 1) \frac{π}{2}}$ .

Unit : Complex variable functions

Q2 (b) Summer-2020 Determine the Mobius transformation which maps $z = \infty, i, 0$ into $w = 0, i, \infty$ .

Unit : Complex variable functions

Q2 (c) Winter-2019 Find the image of the infinite strips $(i) \frac{1}{4} \leq y \leq \frac{1}{2} (i) 0 \leq y \leq \frac{1}{2}$ under the transformation $w = \frac{1}{z}$ . Show the region graphically.

Unit : Complex variable functions

Q3 (b) Winter-2019
Check whether the following functions are analytic or not at any point,

$f (z) = 3 x + y + i (3 y - x)$

$f (z) = z^{\frac{3}{2}}$

Unit : Complex variable functions

Q2 (c( i )) Summer-2020
Find the fourth roots of -1.

Unit : Complex variable functions

Q2 (c ( ii )) Summer-2020
Find the roots of $\log z = i \frac{π}{2}$ .

Unit : Complex variable functions

Q3 (b) Winter-2019
If $f (z) = u + iv$ , is an analytic function,prove that $(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) {|Re f (z)|}^{2} = 2 {|f^{'} (z)|}^{2}$ .

Unit : Complex variable functions

Complex Variables and Partial Differential Equations (3130005)

BE - Semester - Winter - 2019 - 26-11-2019

Total Marks : 70

Q1 (a) Winter-2019 Find the real and imaginary parts of $f (z) = \frac{3 i}{2 + 3 i}$ .

Q1 (a) Summer-2020 Find the analytic function $f (z) = u + iv, if u = x^{3} - 3 {xy}^{2}$

Q1 (b) Summer-2020 Find the roots of the equation, $z^{2} - (5 + i) z + (8 + i) = 0$ .

Q1 (b) Winter-2019 State De-Movire’s formula and hence evaluate ${(1 + i \sqrt{3})}^{100} + {(1 - i \sqrt{3})}^{100} .$

Q1 (c) Winter-2019 Define harmonic function. Show that $u (x, y) = \sinh x \sin y$ is a harmonic function. Find its harmonic conjugate $v (x, y)$ .

Q1 (c ( i )) Summer-2020 Determine & sketch the image $| z | = 1$ of under the transformation $w = z + i$ .

Q1 (c ( ii )) Summer-2020 Find the real and imaginary parts of $f (z) = z^{2} + 3 z$ .

Q2 (a) Winter-2019
Determine the Mobius transformation which maps $z_{1} = 0, z_{2} = 1, z_{3} = \infty$ into $, w_{1} = - 1, w_{2} = - i, w_{3} = 1$ .

Q2 (b) Winter-2019 Define $\log z$ , prove that $i^{i} = e^{- (4 n + 1) \frac{π}{2}}$ .

Q2 (b) Summer-2020 Determine the Mobius transformation which maps $z = \infty, i, 0$ into $w = 0, i, \infty$ .

Q2 (c) Winter-2019 Find the image of the infinite strips $(i) \frac{1}{4} \leq y \leq \frac{1}{2} (i) 0 \leq y \leq \frac{1}{2}$ under the transformation $w = \frac{1}{z}$ . Show the region graphically.

Q3 (b) Winter-2019
Check whether the following functions are analytic or not at any point,

$f (z) = 3 x + y + i (3 y - x)$

$f (z) = z^{\frac{3}{2}}$

Q2 (c( i )) Summer-2020
Find the fourth roots of -1.

Q2 (c ( ii )) Summer-2020
Find the roots of $\log z = i \frac{π}{2}$ .

Q3 (b) Winter-2019
If $f (z) = u + iv$ , is an analytic function,prove that $(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) {|Re f (z)|}^{2} = 2 {|f^{'} (z)|}^{2}$ .

Complex Variables and Partial Differential Equations (3130005)

BE - Semester - Winter - 2019 - 26-11-2019

Total Marks : 70

Q1 (a) Winter-2019 Find the real and imaginary parts of fz=3i 2 + 3i .

Q1 (a) Summer-2020 Find the analytic function fz = u + iv , if u = x3 - 3xy2

Q1 (b) Summer-2020 Find the roots of the equation, z2 - 5+iz + 8+i = 0.

Q1 (b) Winter-2019 State De-Movire’s formula and hence evaluate 1+i 3 100 + 1-i 3 100 .

Q1 (c) Winter-2019 Define harmonic function. Show that ux,y = sinh x sin y is a harmonic function. Find its harmonic conjugate vx,y.

Q1 (c ( i )) Summer-2020 Determine & sketch the image | z | = 1 of under the transformation w = z + i.

Q1 (c ( ii )) Summer-2020 Find the real and imaginary parts of fz = z2 + 3z.

Q2 (a) Winter-2019 Determine the Mobius transformation which maps z1 = 0, z2 = 1, z3 = ∞ into , w1 = -1, w2 = -i, w3 = 1.

Q2 (b) Winter-2019 Define log z, prove that ii = e-4n+1π2 .

Q2 (b) Summer-2020 Determine the Mobius transformation which maps z = ∞, i, 0 into w = 0, i, ∞.

Q2 (c) Winter-2019 Find the image of the infinite strips i 1 4 ≤ y ≤1 2 i 0≤ y ≤1 2 under the transformation w=1 z . Show the region graphically.

Q3 (b) Winter-2019 Check whether the following functions are analytic or not at any point, fz = 3x + y + i3y - x fz = z32

Q2 (c( i )) Summer-2020 Find the fourth roots of -1.

Q2 (c ( ii )) Summer-2020 Find the roots of log z = iπ2.

Q3 (b) Winter-2019 If fz = u + iv , is an analytic function,prove that ∂2∂x2 + ∂2∂y2 Re fz2 = 2 f'z2 .

Q1 (a) Winter-2019 Find the real and imaginary parts of $f (z) = \frac{3 i}{2 + 3 i}$ .

Q1 (a) Summer-2020 Find the analytic function $f (z) = u + iv, if u = x^{3} - 3 {xy}^{2}$

Q1 (b) Summer-2020 Find the roots of the equation, $z^{2} - (5 + i) z + (8 + i) = 0$ .

Q1 (b) Winter-2019 State De-Movire’s formula and hence evaluate ${(1 + i \sqrt{3})}^{100} + {(1 - i \sqrt{3})}^{100} .$

Q1 (c) Winter-2019 Define harmonic function. Show that $u (x, y) = \sinh x \sin y$ is a harmonic function. Find its harmonic conjugate $v (x, y)$ .

Q1 (c ( i )) Summer-2020 Determine & sketch the image $| z | = 1$ of under the transformation $w = z + i$ .

Q1 (c ( ii )) Summer-2020 Find the real and imaginary parts of $f (z) = z^{2} + 3 z$ .

Q2 (a) Winter-2019
Determine the Mobius transformation which maps $z_{1} = 0, z_{2} = 1, z_{3} = \infty$ into $, w_{1} = - 1, w_{2} = - i, w_{3} = 1$ .

Q2 (b) Winter-2019 Define $\log z$ , prove that $i^{i} = e^{- (4 n + 1) \frac{π}{2}}$ .

Q2 (b) Summer-2020 Determine the Mobius transformation which maps $z = \infty, i, 0$ into $w = 0, i, \infty$ .

Q2 (c) Winter-2019 Find the image of the infinite strips $(i) \frac{1}{4} \leq y \leq \frac{1}{2} (i) 0 \leq y \leq \frac{1}{2}$ under the transformation $w = \frac{1}{z}$ . Show the region graphically.

Q3 (b) Winter-2019
Check whether the following functions are analytic or not at any point,

$f (z) = 3 x + y + i (3 y - x)$

$f (z) = z^{\frac{3}{2}}$

Q2 (c( i )) Summer-2020
Find the fourth roots of -1.

Q2 (c ( ii )) Summer-2020
Find the roots of $\log z = i \frac{π}{2}$ .

Q3 (b) Winter-2019
If $f (z) = u + iv$ , is an analytic function,prove that $(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) {|Re f (z)|}^{2} = 2 {|f^{'} (z)|}^{2}$ .