GATE Mechanical Engineering (ME) 2018 Shift 1 Solved Paper

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Question : 28 of 65

Marks: +1, -0

F(z)

is a function of the complex variable

z=x+i y

given by

F(z)=i z+k Re(z)+i Im(z)

For what value of

{k}

will

{f}( {z})

satisfy the Cauchy-Riemann equations?

0
1
$-1$
y

Solution:

F(z)=i . z+k ⋅ Re(z)+i . Im(z)

F(z)=i .(x+i y)+k x+i y

F(z)=(k x-y)+i(x+y)

F(z)=u+i v

Where

u=(k x-y)

and

v=(x+y)

According to Cauchy Riemann Equations,

\;{∂ u}/{∂ x}=\;{∂ v}/{∂ y}

and

\;{∂ u}/{∂ y}=-\;{∂ v}/{∂ x}

From first condition

{k}=1

.

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