Question 1

If $f(z)=e^z$, then $f'(z)$ equals

Accepted Answer

$e^z$

Answer

$ze^z$

Answer

$1$

Answer

$e^{z-1}$

Question 2

A function satisfying Cauchy–Riemann equations with continuous partial derivatives is

Accepted Answer

analytic

Answer

harmonic only

Answer

non-differentiable

Answer

continuous only

Question 3

The singularity of $e^{1/z}$ at $z=0$ is

Accepted Answer

essential

Answer

removable

Answer

regular

Answer

simple pole

Question 4

Using Milne–Thomson method, the analytic function with $u=x^2-y^2$ is

Accepted Answer

$f(z)=z^2$

Answer

$f(z)=e^z$

Answer

$f(z)=z$

Answer

$f(z)=|z|^2$

Question 5

Evaluate $\oint_{|z|=1} \dfrac{z}{z-2}\,dz$

Accepted Answer

$0$

Answer

$-2\pi i$

Answer

$\pi i$

Answer

$2\pi i$

Question 6

Find the residue of $f(z)=\dfrac{1}{z(z-2)}$ at $z=0$

Accepted Answer

$-	frac12$

Answer

$	frac12$

Answer

$1$

Answer

$0$

Question 7

The region of convergence of the Laurent series of $\dfrac{1}{z(z-1)}$ about $z=1$ is

Accepted Answer

$0<|z-1|<1$

Answer

$|z-1|>1$

Answer

$|z-1|<1$

Answer

$|z|<1$

Question 8

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

Accepted Answer

$6\pi i$

Answer

$0$

Answer

$2\pi i$

Answer

$4\pi i$

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

$2x+y$

Answer

$x^2+y^2$

Answer

$xy$

Question 10

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

Accepted Answer

$0$

Answer

$2$

Answer

$-1$

Answer

$1$

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