Complex Analysis Preliminaries

1. Define a group 2. Define an Abelian group 3. Define a field 4. Define a sub-field 5. Define a norm on a field F 6. Define the set of complex numbers 7. Define addition and multiplication in C 8. Define the real and imaginary parts of a complex number 9. Define the modulus of a complex number 10. Define the complex conjugate of a complex number 11. How is the modulus of a complex number and its complex conjugate linked? 12. Describe how we change 1/z into the form a+ib 13. Define a linear vector space 14. Define a normed linear vector space 15. State the Cauchy-Schwartz inequality 16. Define the complex plane and state how we define cos(theta) and sin(theta) in terms of this 17. Are the trigonometric functions injective? 18. Define the argument of a complex number and state what it represents in the complex plane 19. What is the link between the argument of a complex number and the argument of its complex

conjugate? 20. Write a complex number z in terms of its modulus and argument 21. Define the modulus and argument of zw and hence write zw in terms of the moduli and arguments 22. State the sum of angles formulas 23. Write z/w in terms of moduli and arguments 24. Define the modulus and argument of z^n and hence write z^n in terms of moduli and arguments,

what is the name of this formula? 25. Define the n-th root of z, and describe how we find all n of them 26. What are the derivatives of sin(theta) and cos(theta)? 27. What are the Taylor expansions of sin(theta) and cos(theta)?

Metric Spaces 1. Define a metric space 2. Define a convergent sequence 3. Define a Cauchy sequence 4. Define a complete metric space 5. What is the link between convergent sequences and Cauchy sequences in a) a metric space and b) a

complete metric space? 6. Define the open ball Br(x) 7. Define an accumulation point 8. Define an isolated point 9. Define an open set 10. What are the only two subsets of a metric space X which are both open and closed? 11. State whether a) the union of open sets b) the intersection of finitely many open sets is open or

closed 12. Define a closed set 13. Define the closed ball Br(x) 14. Define the boundary of a ball 15. State whether a) the union of finitely many closed sets b) the intersection of closed sets is open or

closed 16. Define the diameter of a subset U of X 17. When is a subset U of X bounded? 18. Define a cover for a subset A of X 19. Define an open cover for a subset A of X

20. Define a compact set K 21. State the Cantor-Intersection lemma 22. State the Heine-Borel theorem 23. Define the limit of a function value 24. When is a function f continuous? 25. When is a function f uniformly continuous? 26. What is the link between continuity and uniform continuity? 27. When is a function f bounded? 28. What can we say about a continuous function f on a compact set K? 29. When are a sequence of functions fj pointwise convergent to f? 30. When are a sequence of functions fj uniformly convergent to f? 31. When are a sequence of functions fj uniformly Cauchy? 32. What can we say about a sequence of uniformly Cauchy functions fj in a subset U of a complete metric

space X? 33. What can we say about the limit f of a sequence of continuous functions fj on a subset U of a metric

space X?

Complex Differentiation 1. When is a function f differentiable at a point z(0) in C? 2. For f,g two differentiable functions and c a complex number, define [cf]’(z0), [f+g]’(z0), [fg]’(z0) and

[f/g]’(z0) 3. When is a function f holomorphic? 4. When is a function f entire? 5. State the Cauchy-Riemann equations 6. How are the Cauchy-Riemann equations for a function f and the concept of holomorphism linked? 7. What is the main example of a function which is not holomorphic in C?

Complex Integration 1. When is a function gamma real differentiable? 2. When is a function gamma a regular path? 3. When is a function gamma a piecewise regular path? 4. Define the speed of a function gamma 5. Define the orientation of a function gamma 6. If gamma is a path, define the path with opposite orientation 7. Define a closed path 8. Define a Jordan path 9. State the important theorem about piecewise regular closed Jordan paths and the complex plane 10. When is a piecewise regular closed Jordan path a) positively oriented and b) negatively oriented? 11. Define the real integral of a function phi 12. Define the path integral for a) a regular path b) a piecewise regular path 13. Expand f(gamma(t))(gamma’(t)) 14. What is the link between the path integral of f over a path and the path with opposite orientation? 15. State the Complex Fundamental Theorem of Calculus 16. When is an open set called pathwise connected? 17. State the two important corollaries about pathwise connected open sets 18. Define a convex set 19. State the important proposition about primitives and convex sets 20. State Cauchy’s Theorem 21. State Cauchy’s Theorem in complex domains 22. State the Cauchy-Goursat Theorem 23. State Cauchy’s Integral Formula 24. Stae the interesting corollary and lemma used to prove Cauchy’s Integral Formula 25. State Cauchy’s Integral Formulae for Derivatives 26. State the estimates on derivatives corollary 27. State Liouville’s Theorem 28. State the Fundamental Theorem of Algebra

Complex Power Series 1. Define a power series in z

2. Define the radius of convergence of a sequence c(n) 3. When is a real series convergent? 4. When is a complex series convergent? 5. When is a complex series absolutely convergent? 6. What is the link between convergence and absolute convergence? 7. If a complex series is convergent, what are the main two other conclusions we can draw? 8. State the Fundamental Theorem of Power Series 9. What is the limit as n tends to infinity of n^(1/n)? 10. State the Weierstrass M-Test 11. State the theorem of uniform convergence and integrals 12. State the theorem of uniform convergence and derivatives 13. What are the two links between the power series and the power series obtained by term-by-term

differentiation? 14. Define the complex exponential function 15. State Euler’s identity 16. Define the Cauchy product of two series 17. Define the principal branch of the logarithm 18. Define sinz, cosz, sinhz, coshz in terms of exponentials 19. State the four links between sin(iz), cos(iz), sinhz, coshz 20. State the two links between sinz, cosz, sinhz, coshz 21. Define sin’(z), cos’(z), sin(z+w), cos(z+w) 22. State (the complex) Taylor’s Theorem and therefore the Taylor expansion of f(z) 23. What is the difference between Taylor’s Theorem in the real case and the complex case? 24. State the Identity Theorem 25. State Proposition 4.7.1 and Lemma 4.7.1 26. Define a connected set 27. State the Isolation of Zeros corollary

Residues 1. State the inversion and double series theorem 2. State the Laurent Expansions theorem 3. Define an isolated singularity 4. Define a removable singularity, a pole and an essential singularity 5. State the important proposition about poles 6. Define a residue 7. Describe the method for finding a residue 8. State the residue theorem