Complex Analysis/Sample Midterm Exam 2

1. Restrict $-\pi <\arg(z)\leq \pi$, and take the corresponding branch of the logarithm:

a.$\log -1-i{\sqrt {3}}$

b.$1^{7+i}$

c. Find all 4 roots of ${\sqrt {2}}+i{\sqrt {2}}$

d $\left|e^{\log(2)+i\log({\sqrt {2}})}\right|$

2. State the Cauchy-Riemann equations for a complex valued function $f(z)$. If you use symbols other then $f$ and $z$ indicate how they relate to these quantities.

3. State whether the give function is holomorphic on the set where it is defined.

a. $\displaystyle 2z+2z^{2}-{\frac {1}{z}}$

b. Let $z=x+iy$ and let $f(z)=e^{ix}$.

c. $zg(z)$ where $g(z)$ satisfies $\displaystyle {\frac {\partial }{\partial {\bar {z}}}}g(z)=0$

d. $|z|$

4. Let $\gamma :[a,b]\to \mathbb {C}$ be a simple closed curve so that $z=0$ lies in the interior of the region bounded by $\gamma$.

a. Suppose $n\geq 0$ and compute

$\oint _{\gamma }z^{n}\,dz,$

simply writing the correct value without any explanation will not receive credit.

b. We now consider the case corresponding to $n=-1$. Please compute

$\oint _{\gamma }z^{-1}\,dz,$

and explain your steps.

c: Now suppose $n\leq -2$ and compute

$\oint _{\gamma }z^{n}\,dz.$

5. Let $\gamma :[0,2\pi ]\to \mathbb {C}$ be given by $\gamma (t)=6e^{it}$. Calculate $\displaystyle \oint _{\gamma }{\frac {\cos \zeta }{\zeta +\pi }}\,d\zeta$

6. Let $u(x,y)=x^{2}-y^{2}$ find a function $v(x,y)$ so that $f=u+iv$ is holomorphic in the complex plane and $v(0,0)=1$.

7.

a. Using the limit characterization of the complex derivative show that ${\bar {z}}$ is not holomorphic.

b. On the other hand show that if ${\frac {\partial }{\partial z}}{\bar {z}}=0$.

c. Do parts (a) and (b) contradict each other, explain why or why not.

8. State Cauchy's integral theorem, and intuitively what you need to know about the function, domain and contour.