An analytic function of a complex variable z=x+i y(i=\sqrt{-1)} is defined as f(z)=x^{2}-y^{2}+i \psi(x, y) where \psi(x, y) is a real function. The value of the imaginary part of f(z) at z=(1+i) is ________________ (round off to 2 decimal places).

GATE Mechanical Engineering (ME) 2020 Shift 1 Solved Paper

Question : 42 of 65

Marks: +1, -0

An analytic function of a complex variable

z=x+i y(i=√{-1)}

is defined as

f(z)=x^{2}-y^{2}+i ψ(x, y)

where

ψ(x, y)

is a real function. The value of the imaginary part of

f(z)

z=(1+i)

is ________________ (round off to 2 decimal places).

Solution:

f(z)=(x^{2}-y^{2})+ξ(x, y)

z=1+i

x=1, y=1

∂ v=\;{∂ v}/{∂ x} d x+\;{∂ v}/{∂ y} d y

Those term of which not containing

x

for analytic

f^{n}

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

⇒ {u}= {x}^{2}- {y}^{2}

2 {x}=\;{∂ {V}}/{∂ {y}} \;{∂ {v}}/{∂ {x}}=(-2 {y})

\;{∂ {V}}/{∂ {y}}=2 {x} \;{∂ {V}}/{∂ {x}}=2 {y}

∂ v=2 y d x+2 {x} ↙{||↙{0}}d y

∂ v=2 y d x

v=2 x y

v(1,1)=2 × 1 × 1

∨(1,1)=2

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