Complex Analysis     MATH 305

Quiz #2

Thursday, May 5, start of class

The second quiz is a theorem and definitions quiz, intended to review some of the most important material from the semester. You should memorize the following theorems, definitions, and formulas. You will be asked to state a particular theorem or definition precisely.

1. Both versions of the triangle inequality (equation (9) in Section 4).
2. Euler's formula (Section 6).
3. The equation of a circle using the modulus (bottom of p. 10).
4. The ε-δ definition of a continuous function (Section 18).
5. The Cauchy-Riemann equations in both rectangular and polar coordinates (Sections 21 and 23).
6. The SCD Theorem on sufficient conditions for differentiability (Section 22).
7. The definition of an analytic function (Section 24).
8. The definition of a harmonic function (Section 26).
9. The principal branch of the logarithmic function (Section 31).
10. The ML Theorem for bounding the modulus of a contour integral (Section 43).
11. The AD Theorem for integration (Section 44).
12. The Cauchy-Goursat Theorem (Section 46).
13. The Cauchy Integral Formula, including its extension (Sections 50 and 51).
14. Liouville's Theorem (Section 53).
15. The Fundamental Theorem of Algebra (Section 53).
16. Taylor's Theorem (Section 57).
17. Laurent's Theorem (Section 60).
18. The residue of a function at an isolated singular point (Section 69).
19. Cauchy's Residue Theorem (Section 70).
20. Three types of isolated singular points: removable singularities, poles, essential singularities (Section 72).