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

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.