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 $-\pi <\arg (z)\le \pi$, 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

${\oint }_{\gamma }\overline{z}{z}^{2}dz$

(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 $u(x,y)=x^3-3x{y}^2-x$, 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 $\frac{\partial f}{\partial z}=u_x+i{v}_x=f_x$.

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

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

(a) $f(z)=\frac{z{e}^z}{z-1}$

(b) $f(z)=e^{|z{|}^2}$

(c) Let $z=x+iy$ and let $f(x+iy)=x^3+x{y}^2+i(x^{2}y+y^3)$

(d) Let $z=r{e}^{i\theta }$ and let $f(z)=r{e}^{-i\theta }$

(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.