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 $$u(x,y)=x^3-3 xy^2-x$$, 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) $$f(z)=e^{|z|^2}$$


 * (c) Let $$z=x+iy$$ 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.