Complex Variables and Partial Differential Equations (3130005)

Question-2a

BE | Semester-3 Summer-2020 | 27-10-2020

Q2) (a)

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

y = 2 x^{2} \Rightarrow dy = 4 x dx

dz = dx + i dy \Rightarrow dz = dx + i 4 x dx \Rightarrow dz = (1 + i 4 x) dx

x : 1 to 2

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

I = \int_{1}^{2} (x^{2} - i 4 x^{4}) (1 + i 4 x) d x

I = \int_{1}^{2} (x^{2} + i 4 x^{3} - i 4 x^{4} - i^{2} 16 x^{5}) d x

I = \int_{1}^{2} (x^{2} + i 4 x^{3} - i 4 x^{4} + 16 x^{5}) d x

I = {[\frac{x^{3}}{3} + i x^{4} - i \frac{4 x^{5}}{5} + \frac{16 x^{6}}{6}]}_{1}^{2}

I = [\frac{8}{3} + i 16 - i \frac{128}{5} + \frac{1024}{6}] - [\frac{1}{3} + i - i \frac{4}{5} + \frac{16}{6}]

I = [\frac{8}{3} + \frac{1024}{6} - \frac{1}{3} - \frac{16}{6}] + i [16 - \frac{128}{5} - 1 + \frac{4}{5}]

I = [\frac{511}{3}] - i [\frac{49}{5}]

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