Complex Analysis/Sample Midterm Exam 1

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

(a) ${\displaystyle \log(1+i{\sqrt {3}})}$
(b) ${\displaystyle (1+i)^{1+i}\!}$
(c) ${\displaystyle \sin(i\pi )\!}$
(d) ${\displaystyle \left|e^{i\pi ^{2}}\right|\!}$

Question 2: Compute the following line integrals:

(a) Let ${\displaystyle \gamma (t)=4e^{2\pi it}}$ for ${\displaystyle t\in [0,1]}$. Compute the line integral
${\displaystyle \oint _{\gamma }{\frac {1}{z^{3}}}\,dz}$
(b) Let ${\displaystyle \gamma (t)=t-it}$ for ${\displaystyle t\in [0,1]}$. Compute the line integral
${\displaystyle \oint _{\gamma }{\bar {z}}z^{2}\,dz}$
(c) Let ${\displaystyle \gamma (t)=e^{it}}$ for ${\displaystyle t\in [0,2\pi ]}$. Compute the line integral
${\displaystyle \oint _{\gamma }e^{z}\cos(z)\,dz}$

Question 3: Let ${\displaystyle u(x,y)=x^{3}-3xy^{2}-x}$, verify that ${\displaystyle u(x,y)}$ is harmonic and find a function ${\displaystyle v(x,y)}$ so that ${\displaystyle v(0,0)=0}$ and ${\displaystyle f=u+iv}$ is a holomorphic function.

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

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

Comment: This problem shows that if ${\displaystyle F}$ and ${\displaystyle f}$ is a function in the complex plane, and ${\displaystyle F(x+i0)=g(x)}$ and ${\displaystyle f(x+i0)=g'(x)}$, then we can use this problem to show that ${\displaystyle \textstyle {\frac {\partial F}{\partial z}}(x+i0)=f(x+i0)}$. 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 ${\displaystyle \textstyle {\frac {\partial F}{\partial z}}(z)=f(z)}$. (For example, take ${\displaystyle F(z)=\sin(z)}$ and ${\displaystyle f(z)=\cos(z)}$ then ${\displaystyle g(x)=\sin(x)}$, so it must be that ${\displaystyle \textstyle {\frac {\partial F}{\partial z}}=f(z)=\cos(z)}$.)

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

(a) ${\displaystyle f(z)={\frac {ze^{z}}{z-1}}}$
(b) ${\displaystyle f(z)=e^{|z|^{2}}}$
(c) Let ${\displaystyle z=x+iy}$ and let ${\displaystyle f(x+iy)=x^{3}+xy^{2}+i(x^{2}y+y^{3})}$
(d) Let ${\displaystyle z=re^{i\theta }}$ and let ${\displaystyle f(z)=re^{-i\theta }}$
(e) Let ${\displaystyle z=x+iy}$ and let ${\displaystyle f(z)=e^{ix}}$

Question 7: State 4 ways to test if a function ${\displaystyle f(z)}$is holomorphic.