Complex Analysis/Sample Midterm Exam 1

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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

\gamma(t)=e^{it}

for

t \in [0, 2\pi]

. Compute the line integral

\oint_\gamma e^z\cos(z)\,dz

Question 3: Let $,u(x,y)=x^3-xy^2-x$ , 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}

z=x+iy

and let

f(x+iy)=x^3+xy^2+i(x^2y+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.

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