Complex Variables and Partial Differential Equations (3130005)

Question-4a

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

Q4) (a)

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

Here,

z = t + i t^{2} - - - - - (1)

\Rightarrow x + i y = t + i t^{2}

\Rightarrow x = t & y = t^{2}

\Rightarrow dx = dt & dy = 2 t dt

Now,

dz = dx + i dy

\Rightarrow dz = dt + i 2 t dt

\Rightarrow dz = (1 + i 2 t) dt

Limit of Integration

If

z = 0

then by

[1], t = 0

If

z = 2 + 4 i

then by

[1], t = 2

\Rightarrow t : 0 to 2

\int_{0}^{2 + 4 i} Re (z) d z = \int_{0}^{2 + 4 i} x d z = \int_{0}^{2} t (1 + i 2 t) dt = \int_{0}^{2} (t + i 2 t^{2}) dt = {[\frac{t^{2}}{2} + i \frac{2 t^{3}}{3}]}_{0}^{2} = 2 + i \frac{16}{3}

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