# GATE Mechanical 2014 Set 2 | Question: 26

An analytic function of a complex variable $z=x+iy$ is expressed as $f(z)=u(x,y)+iv(x,y)$ , where $i=\sqrt{-1}$ If $u(x,y)=2xy$, then $v(x,y)$ must be

1. $x^2$+$y^2$+constant
2. $x^2$-$y^2$+constant
3. -$x^2$+$y^2$+constant
4. -$x^2$-$y^2$+constant
in Calculus
recategorized

## Related questions

If $z$ is a complex variable, the value of $\displaystyle{} \int_{5}^{3i}\dfrac{dz}{z}$ is $− 0.511−1.57i$ $− 0.511+1.57i$ $0.511− 1.57i$ $0.511+1.57i$
An analytic function of a complex variable $z = x + i y$ is expressed as $f (z) = u(x, y) + i v(x, y)$ , where $i=\sqrt{-1}$ . If $u(x, y) = x^2 − y^2$ , then expression for $v(x, y)$ in terms of $x$, $y$ and a general constant $c$ would be $xy + c \\$ $\dfrac{x^2+y^2}{2}+c \\$ $2xy+c \\$ $\dfrac{(x-y)^2}{2}+c$
Let $z$ be a complex variable. For a counter-clockwise integration around a unit circle $C$, centered at origin, $\oint_C \frac{1}{5z-4} dz=A \pi i$, the value of $A$ is $2/5$ $1/2$ $2$ $4/5$
Curl of vector $\overrightarrow{F}=x^2z^2\hat{i}-2xy^2z\hat{j}+2y^2z^3\hat{k}$ is $(4yz^3+2xy^2)\hat{i}+2x^2z\hat{j}-2y^2z\hat{k}$ $(4yz^3+2xy^2)\hat{i}-2x^2z\hat{j}-2y^2z\hat{k}$ $2xz^2\hat{i}-4xyz\hat{j}+6y^2z^2\hat{k}$ $2xz^2\hat{i}+4xyz\hat{j}+6y^2z^2\hat{k}$
$\displaystyle{}\lim_{x\rightarrow 0} \left( \dfrac{e^{2x}-1}{\sin(4x)} \right )$ is equal to $0$ $0.5$ $1$ $2$