Complex Analysis/Sample Midterm Exam 2

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

\log -1-i\sqrt{3}

1^{7+i}

c. Find all 4 roots of

\sqrt{2}+i\sqrt{2}

\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.

\displaystyle 2z+2z^2-\frac{1}{z}

b. Let

z=x+iy

and let

f(z)=e^{ix}

zg(z)

where

g(z)

satisfies

\displaystyle \frac{\partial}{\partial \bar z} g(z)=0

|z|

4. Let $\gamma:[a,b]\to \C$ 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 $\gamma:[0,2\pi]\to \C$ 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 $v(0,0)=1$ .

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.

This article is issued from Wikiversity - version of the Saturday, November 19, 2011. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.