Complex Analysis.docx

Complex Analysis

Preliminaries1. How is the modulus of a complex number and its complex conjugate linked?2. Define the complex plane and state how we define cos(theta) and sin(theta) in terms of this3. Are the trigonometric functions injective?4. Define the argument of a complex number and state what it represents in the complex plane5. What is the link between the argument of a complex number and the argument of its complex conjugate?6. Write a complex number z in terms of its modulus and argument in both exponential and trigonometric forms7. Write zw in terms of the moduli and arguments8. Write z/w in terms of moduli and arguments9. Write z^n in terms of moduli and arguments, what is the name of this formula?10. Define the n-th root of z, and describe how we find all n of them11. Define the geometric series and state conditions of convergence and its limit12. Define the limsup of a sequence13. State LHopitals ruleMetric Spaces1. Define an accumulation point2. Define an isolated point3. Define an open set4. Define a closed set5. Define the diameter of a subset U of X6. When is a subset U of X bounded?Complex Differentiation1. For f,g two differentiable functions and c a complex number, define [cf](z0), [f+g](z0), [fg](z0) and [f/g](z0)2. When is a function f holomorphic?3. When is a function f entire?4. State the Cauchy-Riemann equations5. How are the Cauchy-Riemann equations for a function f and the concept of holomorphism linked?Complex Integration1. When is a function gamma a regular path?2. When is a function gamma a piecewise regular path?3. Define the speed of a path4. Define the length of a path5. Define the orientation of a path6. If gamma is a path, define the path with opposite orientation7. Define a closed path8. Define a Jordan path9. When is a piecewise regular closed Jordan path a) positively oriented and b) negatively oriented?10. Define the path integral for a) a regular path b) a piecewise regular path11. What is the link between the path integral of f over a path and the path with opposite orientation?12. State the Complex Fundamental Theorem of Calculus13. When is an open set called pathwise connected?14. State the two important corollaries about pathwise connected open sets15. State Cauchys Theorem16. State Cauchys Theorem in complex domains17. State Cauchys Integral Formula18. State Cauchys Integral Formulae for Derivatives19. State Liouvilles Theorem20. State the Fundamental Theorem of AlgebraComplex Power Series1. Define a power series in z2. Define the radius of convergence of a sequence c(n)3. State the Fundamental Theorem of Power Series4. What is the limit as n tends to infinity of n^(1/n)?5. State the differentiation of power series theorem6. Define the complex exponential function7. State Eulers identity8. Define sinz, cosz, sinhz, coshz in terms of exponentials9. State the four links between sin(iz), cos(iz), sinhz, coshz10. State the two links between sinz, cosz, sinhz, coshz11. Define sin(z), cos(z), sin(z+w), cos(z+w)12. State (the complex) Taylors Theorem and therefore the Taylor expansion of f(z)13. State the Taylor expansions of sinz and cosz14. State the Identity Theorem15. Define a connected set16. State the isolation of zeros corollaryResidues1. State the Laurent expansions theorem2. Define an isolated singularity3. Define a removable singularity, a pole and an essential singularity4. Define a simple pole5. Define a residue6. Describe some methods for finding residues7. State the residue theorem