# Complex Analysis/Sample Midterm Exam 2

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

a.${\displaystyle \log -1-i{\sqrt {3}}}$
b.${\displaystyle 1^{7+i}}$
c. Find all 4 roots of ${\displaystyle {\sqrt {2}}+i{\sqrt {2}}}$
d ${\displaystyle \left|e^{\log(2)+i\log({\sqrt {2}})}\right|}$

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

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

a. ${\displaystyle \displaystyle 2z+2z^{2}-{\frac {1}{z}}}$
b. Let ${\displaystyle z=x+iy}$ and let ${\displaystyle f(z)=e^{ix}}$.
c. ${\displaystyle zg(z)}$ where ${\displaystyle g(z)}$ satisfies ${\displaystyle \displaystyle {\frac {\partial }{\partial {\bar {z}}}}g(z)=0}$
d. ${\displaystyle |z|}$

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

a. Suppose ${\displaystyle n\geq 0}$ and compute
${\displaystyle \oint _{\gamma }z^{n}\,dz,}$
simply writing the correct value without any explanation will not receive credit.
b. We now consider the case corresponding to ${\displaystyle n=-1}$. Please compute
${\displaystyle \oint _{\gamma }z^{-1}\,dz,}$
c: Now suppose ${\displaystyle n\leq -2}$ and compute
${\displaystyle \oint _{\gamma }z^{n}\,dz.}$

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

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

7.

a. Using the limit characterization of the complex derivative show that ${\displaystyle {\bar {z}}}$ is not holomorphic.
b. On the other hand show that if ${\displaystyle {\frac {\partial }{\partial z}}{\bar {z}}=0}$.
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.