GATE Papers >> Mechanical >> 2018 >> Question No 14

Question No. 14 Mechanical | GATE 2018

$ F\left(z\right) $ is a function of the complex variable $ z=x+iy $ given by

                              $ F\left(z\right)=i\;z\;+\;k\;Re\;\left(z\right)+i\;Im\left(z\right) $.

For what value of k will $ F\left(z\right) $ satisfy the Cauchy-Riemann equations?


Answer : (B) 1


Solution of Question No 14 of GATE 2018 Mechanical Paper

$ \begin{array}{l}\mathrm Z=\mathrm x+\mathrm{iy}\\\mathrm F(\mathrm z)=\mathrm{iz}+\mathrm k\;\mathrm{Re}(\mathrm z)+{\mathrm{iI}}_\mathrm m(\mathrm z)\\\mathrm{Re}\;(\mathrm z)=\mathrm x,\;{\mathrm I}_\mathrm m(\mathrm z)=\mathrm y\\\downarrow\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\downarrow\\\mathrm{Real}\;\mathrm{part}\;\;\;\;\;\;\;\mathrm{Imaginary}\;\mathrm{part}\\\mathrm F(\mathrm z)=\mathrm i(\mathrm x+\mathrm{iy})+\mathrm{kx}+\mathrm{iy}=\mathrm{ix}+\mathrm i^2\mathrm y+\mathrm{kx}+\mathrm{iy}\\\mathrm F(\mathrm z)=\mathrm{ix}+\mathrm{iy}+\mathrm{kx}-\mathrm y=\mathrm{kx}-\mathrm y+\mathrm i(\mathrm x+\mathrm y)\\\mathrm F(\mathrm z)=\mathrm u+\mathrm{iv}\\\mathrm u=\mathrm{kx}-\mathrm y,\;\;\;\;\mathrm V=\mathrm x+\mathrm y\end{array} $

Cauchy - Riemann equation are

$ \begin{array}{l}\frac{\partial\mathrm u}{\partial\mathrm x}=\frac{\displaystyle\partial\mathrm v}{\displaystyle\partial\mathrm y},\frac{\displaystyle\partial\mathrm v}{\displaystyle\partial\mathrm x}=\frac{\displaystyle\partial\mathrm u}{\displaystyle\partial\mathrm y}\\\frac{\displaystyle\partial\mathrm u}{\displaystyle\partial\mathrm x}=\frac{\displaystyle\partial\mathrm v}{\displaystyle\partial\mathrm y}\\\frac{\displaystyle\partial}{\displaystyle\partial\mathrm x}(\mathrm{kx}-\mathrm y)=\frac{\displaystyle\partial}{\displaystyle\partial\mathrm y}(\mathrm x+\mathrm y)\\\mathrm k=1\end{array} $

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