Question 1

If $f(z)=z^3$, then $f'(z)$ is

Accepted Answer

$3z^2$

Answer

$3z$

Answer

$z^2$

Answer

$z^3$

Question 2

If $u(x,y)=x^2-y^2$, then $u$ is harmonic if

Accepted Answer

$u_{xx}+u_{yy}=0$

Answer

$u_x=u_y$

Answer

$u_x+u_y=0$

Answer

$u_{xx}=u_{yy}$

Question 3

The singularity of $\dfrac{1}{z^2}$ at $z=0$ is

Accepted Answer

pole of order $2$

Answer

essential

Answer

removable

Answer

simple pole

Question 4

Find the harmonic conjugate $v$ of $u=x^3-3xy^2$

Accepted Answer

$v=3x^2y-y^3+C$

Answer

$v=x^2y$

Answer

$v=3xy^2-x^3$

Answer

$v=x^3+y^3$

Question 5

Using Cauchy Integral Formula, evaluate $\oint_{|z|=2} \dfrac{z}{z-1}\,dz$

Accepted Answer

$2\pi i$

Answer

$\pi i$

Answer

$4\pi i$

Answer

$0$

Question 6

The Laurent series of $\dfrac{1}{z(z-2)}$ about $z=0$ converges in

Accepted Answer

$|z|<2$

Answer

$|z|<1$

Answer

$|z|>1$

Answer

$|z|>2$

Question 7

Find the residue of $f(z)=\dfrac{1}{z(z-1)}$ at $z=0$

Accepted Answer

$-1$

Answer

$-2$

Answer

$0$

Answer

$1$

Question 8

Evaluate $\oint_{|z|=3} \dfrac{z^2}{(z-1)(z-2)}\,dz$

Accepted Answer

$6\pi i$

Answer

$4\pi i$

Answer

$0$

Answer

$2\pi i$

Question 9

If $f(z)=u+iv$ is analytic and $u=x^2-y^2+2x$, then $v$ equals

Accepted Answer

$2xy+2y+C$

Answer

$xy$

Answer

$2x-y$

Answer

$x^2+y^2$

Question 10

Find the coefficient of $\dfrac{1}{z}$ in the Laurent expansion of $\dfrac{z^2}{z^2-1}$ about $z=0$

Accepted Answer

$0$

Answer

$2$

Answer

$1$

Answer

$-1$

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