Review for Exam 1
Our first exam is next Wednesday, September 30. Here are some topics and problems to study.
Definitions and theorems
- Abs(z) the absolute value of \(z\)
- arg(z) the set of arguments of \(z\)
- Arg(z) the principal argument of \(z\)
- \(w^{1/n}\) the principal \(n^{\text{th}}\) root of \(w\)
- Limit the \(\varepsilon\text{-}\delta\) definition
- \(f'(z_0)\) the derivative of a function at a point
- Function that’s analytic at a point \(z_0\) and/or analytic on a region
Problems
Find both square roots of \(1+i\). Identify which one is the principal square root.
Separate \(f(z)=1+z+z^3\) into its real and imaginary parts.
Writing \(f(x+i y)=u(x,y)+i v(x,y)\), where \(x,y\in \mathbb{R}\) and \(u,v\) map \(\mathbb{R}^2\to \mathbb{R}\), use the Cauchy-Riemann equations to determine which of the following defines an analytic function.
\(f(x+i y)=\left(x^2-y^2+e^x\cos (y)\right)+i\left(2x y+e^x\sin (y)\right)\)
\(f(x+i y)=(x+y)+i(x-y)\)
\(f(z)=\bar{z}\)
Use the definition of the derivative to show that the function \(f(z)=\bar{z}\) is nowhere differentiable.
Express the following complex numbers in the form \(a+b i\).
\(1/(1+i)\)
\(\displaystyle \left(\frac{1+i}{\sqrt{2}}\right)^{100}\)
Let \(R=\{z\in \mathbb{C}:1<|z|<2,\ 0\leq \arg(z)<\pi \}\) and let \(R^2\) denote the image of \(R\) under the square function.
Sketch \(R\) in the plane. Be sure to indicate any edges not contained in \(R\) with dashed lines, while edges that are contained in \(R\) should be solid.
Is \(R\) open, closed, or neither? You needn’t prove or even justify your assertion.
Sketch \(R^2\) in the plane. Be sure to indicate the image of each edge of \(R\).
Is \(R^2\) open, closed, or neither? You needn’t prove or even justify your assertion.
Recall that the cross-ratio of four complex numbers \(z,z_1,z_2,z_3\) is defined by \[ \left[z,z_1,z_2,z_3\right] =\frac{\left(z-z_1\right)\left(z_2-z_3\right)} {\left(z-z_3\right)\left(z_2-z_1\right)}. \]
If \(T(z)=\left[z,z_1,z_2,z_3\right]\), then what are the images of \(z_1\), \(z_2\), and \(z_3\) under \(T\)?
Use the cross-ratio to find a Möbius transformation that sends \[2\to0, \: 4\to1, \text{ and } 3+i \to \infty.\]
What is the image of the circle of radius 1 centered at 3 under the action of your Möbius transformation?
Write down an \(\varepsilon\text{-}\delta\) proof of the fact that \[ \lim_{z \to i} \left((1+i)z + 1\right) = i. \]
Let \[f(z) = \frac{(1+i) z^2}{2 z+1}.\] We consider \(f\) as a map of the Riemann sphere to itself that fixes \(\infty\).
Compute \(f'(\infty)\) and interpret the result.
Questions
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