Midterm 2 study guide

Things you should know:

1. State equivalent statements of the meaning of continuous function.
2. Explain how if \(f\) is a continuous function and \(U\) is an open set, then \(f^{-1}(U)\) is open.
3. Show using sequences that if \(K\) is closed and \(f\) is continuous then \(f^{-1}(K)\) is closed.
4. Definition of (sequentially) compact sets.
5. Prove that a compact set is closed and bounded.
6. Prove that if \(K\) is a compact set and \(f\) is continuous, then \(f(K)\) is a compact set.
7. Prove that if \(K\) is compact and \(F\) is closed then \(K\cap F\) is compact.
8. Test if a polynomial is analytic.
9. Find the imaginary part of an analytic polynomial whose real part you know, and vice versa, and write as a polynomial in \(z\).
10. Definition of radius of convergence for a power series.
11. Prove the “uniqueness theorem” (2.12).
12. Definition of complex-differentiable function.
13. Prove that a complex-differentiable function must satisfy the Cauchy-Riemann equations.
14. State the converse: When does satisfiability of the Cauchy-Riemann equations imply the function is complex differentiable?
15. Sketch the proof of the previous part.
16. Work with the Cauchy-Riemann equations to determine if a given real/imaginary part can be extended to an analytic function.
17. Definition and main properties of \(e^z\).