Complex Variables and Partial Differential Equations (3130005)

Question-3a

BE | Semester-3 Winter-2019 | 26-11-2019

Q3) (a)

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

The equation of straight line from 0=(0,0) to 1+i=(1,1) is

|\begin{matrix} x & y & 1 \\ 1 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}| = 0

⟹x(1-0)-y(1-0)+1(0+0)=0⟹x-y=0⟹y=x⟹dy=dx

dz=dx+i dy⟹dz=dx+idx⟹dz=(1+i ) dx

x:0 to 1

I = \int_{C} (x - y + i x^{2}) d z

\Rightarrow I = \int_{0}^{1} (x - x + i x^{2}) (1 + i) d x

\Rightarrow I = i (1 + i) \int_{0}^{1} x^{2} d x = (i - 1) {[\frac{x^{3}}{3}]}_{0}^{1} = \frac{i - 1}{3}

Questions

Go to Question Paper