The function f(z) of complex variable z=x+i y, where i=\sqrt{-1}, is given as f(z)=(x^{3}- 3 x y^{2})+i v(x, y). For this function to be analytic, v(x, y) should be

Correct Answer is : (3x2y−y3)+ constant

(3 x^{2} y^{2}-y^{3})+ constant

GATE Mechanical Engineering (ME) 2020 Shift 2 Solved Paper

Question : 59 of 65

Marks: +1, -0

The function

f(z)

of complex variable

z=x+i y

, where

i=√{-1}

, is given as

f(z)=(x^{3}-

3 x y^{2})+i v(x, y)

. For this function to be analytic,

v(x, y)

should be

Solution:

f(z)=(x^{3}-3 x y^{2})+i v(x, y)

\;{∂ u}/{∂ x}=\;{∂ v}/{∂ y} ; \;{∂ u}/{∂ y}=-\;{∂ v}/{∂ x}

\;{∂ u}/{∂ x}=\;{∂ v}/{∂ y}

3 x^{2}-3 y^{2}=\;{∂ v}/{∂ y}

∫ ∂ v=∫(3 x^{2}-3 y^{2}) ∂ y+

constant

v=3 x^{2} y-y^{3}+

constant

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