Complex Variables and Partial Differential Equations (3130005)

CVPDE-3130005

Complex Variable - Integration

BE | Semester 3 Unit : Complex Variable - Integration

Q2 (a) Summer-2020
Evaluate $\int_{C} (x^{2} - i y^{2}) d z$ along the parabola $y = 2 x^{2}$ from $(1, 2)$ to $(2, 8)$ .

Unit : Complex Variable - Integration

Q2 (c(i)) Summer-2020
Evaluate $\int_{C} \frac{e^{- z}}{z + 1} d z$ ,where $C : |z| = \frac{1}{2}$ .

Unit : Complex Variable - Integration

Q2 (c(ii)) Summer-2020
Find the radius of convergence of $\sum_{1}^{\infty} {(1 + \frac{1}{n})}^{n^{2}} z^{n}$ .

Unit : Complex Variable - Integration

Q3 (a) Winter-2019
Evaluate $\int_{C} (x - y + i x^{2}) d z$ ; Where $C$ is a straight line from $z = 0 to z = 1 + i$ .

Unit : Complex Variable - Integration

Q3 (a) Summer-2020
Evaluate $\int_{C} \frac{1}{z^{2}} d z$ , Where $C : |z| = 1$ .

Unit : Complex Variable - Integration

Q3 (c ( i )) Winter-2019
Evaluate the following:

$(i) \int_{C} \frac{z + 3}{z - 1} d z$ ; Where C is the circle $(a) |z| = 2 (b) |z| = \frac{1}{2}$ .

Unit : Complex Variable - Integration

Q3 (c ( ii )) Winter-2019 $\int_{C} \frac{\sin z}{{(z - \frac{π}{4})}^{3}} d z$ ; where C is the circle $|z| = 1$ .

Unit : Complex Variable - Integration

Q3 (a) Summer-2020
Evaluate $\int_{C} \frac{z + 4}{z^{2} + 2 z + 5} d z$ , Where $C : |z + 1| = 1$ .

Unit : Complex Variable - Integration

Q4 (a) Winter-2019 Evaluate $\int_{0}^{2 + 4 i} Re (z) d z$ along the curve $z = t + i t^{2}$ .

Unit : Complex Variable - Integration

Complex Variables and Partial Differential Equations (3130005)

BE - Semester - Summer - 2020 - 27-10-2020

Total Marks : 70

Q2 (a) Summer-2020
Evaluate $\int_{C} (x^{2} - i y^{2}) d z$ along the parabola $y = 2 x^{2}$ from $(1, 2)$ to $(2, 8)$ .

Q2 (c(i)) Summer-2020
Evaluate $\int_{C} \frac{e^{- z}}{z + 1} d z$ ,where $C : |z| = \frac{1}{2}$ .

Q2 (c(ii)) Summer-2020
Find the radius of convergence of $\sum_{1}^{\infty} {(1 + \frac{1}{n})}^{n^{2}} z^{n}$ .

Q3 (a) Winter-2019
Evaluate $\int_{C} (x - y + i x^{2}) d z$ ; Where $C$ is a straight line from $z = 0 to z = 1 + i$ .

Q3 (a) Summer-2020
Evaluate $\int_{C} \frac{1}{z^{2}} d z$ , Where $C : |z| = 1$ .

Q3 (c ( i )) Winter-2019
Evaluate the following:

$(i) \int_{C} \frac{z + 3}{z - 1} d z$ ; Where C is the circle $(a) |z| = 2 (b) |z| = \frac{1}{2}$ .

Q3 (c ( ii )) Winter-2019 $\int_{C} \frac{\sin z}{{(z - \frac{π}{4})}^{3}} d z$ ; where C is the circle $|z| = 1$ .

Q3 (a) Summer-2020
Evaluate $\int_{C} \frac{z + 4}{z^{2} + 2 z + 5} d z$ , Where $C : |z + 1| = 1$ .

Q4 (a) Winter-2019 Evaluate $\int_{0}^{2 + 4 i} Re (z) d z$ along the curve $z = t + i t^{2}$ .

Complex Variables and Partial Differential Equations (3130005)

BE - Semester - Summer - 2020 - 27-10-2020

Total Marks : 70

Q2 (a) Summer-2020 Evaluate∫Cx2 - i y2 dz along the parabola y = 2x2 from 1, 2 to 2, 8.

Q2 (c(i)) Summer-2020 Evaluate ∫C e-z z+1 dz ,where C: z = 12.

Q2 (c(ii)) Summer-2020 Find the radius of convergence of ∑1∞ 1+1nn2 zn .

Q3 (a) Winter-2019 Evaluate ∫Cx-y+i x2 dz ; Where C is a straight line from z = 0 to z = 1 + i .

Q3 (a) Summer-2020 Evaluate ∫C 1z2 dz , Where C : z = 1.

Q3 (c ( i )) Winter-2019 Evaluate the following: i ∫C z + 3 z - 1dz ; Where C is the circle a z = 2 b z = 1 2 .

Q3 (c ( ii )) Winter-2019 ∫C sin z z - π 43 dz ; where C is the circle z = 1.

Q3 (a) Summer-2020 Evaluate ∫C z+4z2+2z+5 dz , Where C : z+1 = 1.

Q4 (a) Winter-2019 Evaluate ∫02+4iRez dz along the curve z = t + i t2.

Q2 (a) Summer-2020
Evaluate $\int_{C} (x^{2} - i y^{2}) d z$ along the parabola $y = 2 x^{2}$ from $(1, 2)$ to $(2, 8)$ .

Q2 (c(i)) Summer-2020
Evaluate $\int_{C} \frac{e^{- z}}{z + 1} d z$ ,where $C : |z| = \frac{1}{2}$ .

Q2 (c(ii)) Summer-2020
Find the radius of convergence of $\sum_{1}^{\infty} {(1 + \frac{1}{n})}^{n^{2}} z^{n}$ .

Q3 (a) Winter-2019
Evaluate $\int_{C} (x - y + i x^{2}) d z$ ; Where $C$ is a straight line from $z = 0 to z = 1 + i$ .

Q3 (a) Summer-2020
Evaluate $\int_{C} \frac{1}{z^{2}} d z$ , Where $C : |z| = 1$ .

Q3 (c ( i )) Winter-2019
Evaluate the following:

$(i) \int_{C} \frac{z + 3}{z - 1} d z$ ; Where C is the circle $(a) |z| = 2 (b) |z| = \frac{1}{2}$ .

Q3 (c ( ii )) Winter-2019 $\int_{C} \frac{\sin z}{{(z - \frac{π}{4})}^{3}} d z$ ; where C is the circle $|z| = 1$ .

Q3 (a) Summer-2020
Evaluate $\int_{C} \frac{z + 4}{z^{2} + 2 z + 5} d z$ , Where $C : |z + 1| = 1$ .

Q4 (a) Winter-2019 Evaluate $\int_{0}^{2 + 4 i} Re (z) d z$ along the curve $z = t + i t^{2}$ .