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GATE IN 2019 Official Paper

Option 3 : f(z) = constant

CT 1: Determinants and Matrices

3135

10 Questions
10 Marks
12 Mins

**Explanation:**

f(z) = u(x, y) + i v(x, y)

f*(z) = u (x, y) - i v(x, y)

f(z) is analytic function

⇒ u_{x} = v_{y} and u_{y} = - v_{x}

\(\Rightarrow \frac{{\partial u}}{{\partial x}} = \frac{{\partial v}}{{\partial y}}~~and~~\frac{{\partial u}}{{\partial y}} = - \frac{{\partial v}}{{\partial x}}\) ----(1)

f*(z) is analytic function

\(\Rightarrow \frac{{\partial u}}{{\partial x}} = - \frac{{\partial v}}{{\partial y}}~~and~~\frac{{\partial u}}{{\partial y}} = \frac{{\partial v}}{{\partial x}}\) ----(2)

From (1) and (2)

\(\Rightarrow \frac{{\partial u}}{{\partial x}} = - \frac{{\partial u}}{{\partial x}}\) ⇒ u is constant

⇒ v is constant

f(z) = u + iv