Complex Variables and Partial Differential Equations (3130005)

Question-1a

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

Q1) (a)

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

Here,

u (x, y) = x^{3} - 3 {xy}^{2}

\Rightarrow \frac{\partial u}{\partial x} = 3 x^{2} - 3 y^{2} & \frac{\partial u}{\partial y} = - 6 xy - - - - - (1)

Let,

f (z) = u + i v

\Rightarrow f^{'} (z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}

\Rightarrow f^{'} (z) = \frac{\partial u}{\partial x} - i \frac{\partial v}{\partial y} (∵ CR Equation)

\Rightarrow f^{'} (z) = (3 x^{2} - 3 y^{2}) - i (- 6 xy) (∵ 1)

\Rightarrow f^{'} (z) = 3 z^{2} (∵ x = z & y = 0)

\Rightarrow f (z) = 3 \int z^{2} dz (∵ Taking Integration)

\Rightarrow f (z) = z^{3} + c

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