# Complex Analysis/Sample Midterm Exam 1

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

(a) $\log(1+i{\sqrt {3}})$ (b) $(1+i)^{1+i}\!$ (c) $\sin(i\pi )\!$ (d) $\left|e^{i\pi ^{2}}\right|\!$ Question 2: Compute the following line integrals:

(a) Let $\gamma (t)=4e^{2\pi it}$ for $t\in [0,1]$ . Compute the line integral
$\oint _{\gamma }{\frac {1}{z^{3}}}\,dz$ (b) Let $\gamma (t)=t-it$ for $t\in [0,1]$ . Compute the line integral
$\oint _{\gamma }{\bar {z}}z^{2}\,dz$ (c) Let $\gamma (t)=e^{it}$ for $t\in [0,2\pi ]$ . Compute the line integral
$\oint _{\gamma }e^{z}\cos(z)\,dz$ Question 3: Let $u(x,y)=x^{3}-3xy^{2}-x$ , verify that $u(x,y)$ is harmonic and find a function $v(x,y)$ so that $v(0,0)=0$ and $f=u+iv$ is a holomorphic function.

Question 4: Explain why there is no complex number $z$ so that $e^{z}=0$ .

Question 5: We often use the formulas from ordinary calculus to compute complex derivatives. This problem is part of the justification. Show that if $f=u+iv$ is holomorphic then ${\frac {\partial f}{\partial z}}=u_{x}+iv_{x}=f_{x}$ .

Comment: This problem shows that if $F$ and $f$ is a function in the complex plane, and $F(x+i0)=g(x)$ and $f(x+i0)=g'(x)$ , then we can use this problem to show that $\textstyle {\frac {\partial F}{\partial z}}(x+i0)=f(x+i0)$ . We will see later that if two holomorphic functions agree on a line then they agree everywhere. So it would have to be the case that $\textstyle {\frac {\partial F}{\partial z}}(z)=f(z)$ . (For example, take $F(z)=\sin(z)$ and $f(z)=\cos(z)$ then $g(x)=\sin(x)$ , so it must be that $\textstyle {\frac {\partial F}{\partial z}}=f(z)=\cos(z)$ .)

Question 6: Decide whether or not the following functions are holomorphic where they are defined.

(a) $f(z)={\frac {ze^{z}}{z-1}}$ (b) $f(z)=e^{|z|^{2}}$ (c) Let $z=x+iy$ and let $f(x+iy)=x^{3}+xy^{2}+i(x^{2}y+y^{3})$ (d) Let $z=re^{i\theta }$ and let $f(z)=re^{-i\theta }$ (e) Let $z=x+iy$ and let $f(z)=e^{ix}$ Question 7: State 4 ways to test if a function $f(z)$ is holomorphic.