Complex Analysis/Sample Midterm Exam 1

This exam has a total of 100 points. You have 50 minutes. Partial credit will be awarded so showing your work can only help your grade

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 $\textstyle {\frac {\partial F}{\partial z}}=f(z)=\cos(z)$ .)