Complex Analysis     MATH 305

Final Exam

Tuesday, Dec. 13, 3:00 - 5:30 pm, Swords 302

The Final Exam is cumulative, covering all the material we have discussed in Chapters 1 - 7 of the course text. This is everything except for Sections 11, 27, 28, 35, 36, 47, 58, 61, 67, 71, 75, 76, 77, 80 and onwards. You should go over all homework assignments (including the partial solutions for HW #1 and #6 on Moodle), computer lab #1, the textbook and your class notes.

We will review for the exam on Monday, Dec. 12, 1:00 - 2:30 pm in Swords 302. Please come prepared with specific questions. Some sample final exam questions are available here (PDF file).

Note: You will be allowed one "cheat sheet" 8.5 x 11 piece of paper, front and back, full of whatever formulas, theorems, etc. you wish. Also, no calculators are allowed on the exam so be prepared to answer questions without your personal calculator.

The following topics, definitions and theorems are important material for the exam. You may be asked to define some terms precisely as well as state and/or prove important facts and theorems. Keep in mind that many of the problems in the text provide answers or hints, either as part of the question or listed after the problem. You will not have such an advantage on the exam.

  1. Complex Numbers: complex plane, addition, multiplication, basic algebraic properties, multiplicative and additive inverses, conjugate

  2. Vectors and Moduli: addition and subtraction of complex numbers vectorially, modulus of a complex number, equation of a circle using the modulus, triangle inequalities, properties of the modulus (e.g., the modulus of a product equals the product of the moduli)

  3. Complex Conjugate: properties of, connection with modulus

  4. Exponential Form: polar coordinates, Euler's formula, argument versus Argument, products and powers in exponential form, de Moivre's formula, arguments of products and quotients

  5. Roots of Complex Numbers: how to find the nth roots of a complex number, geometric description of the location of the roots, roots of unity

  6. Functions: writing functions in terms of real and imaginary parts, mappings from the z-plane to the w-plane, mapping the exponential function

  7. Limits: epsilon-delta definition, approaching from different directions, CR limit theorem, Big limit theorem (BLT), limits involving infinity, the Riemann sphere, LIPI theorem, definition of a continuous function

  8. Derivatives and Analytic Functions: definition of the derivative, basic differentiation formulas, Cauchy-Riemann equations (in both rectangular and polar coordinates), SCD theorem, definition of an analytic function, entire functions, singular points, finding where a function is analytic (if at all)

  9. Harmonic Functions: Laplace's equation, definition of a harmonic function, the harmonic conjugate, finding harmonic conjugates

  10. Important Functions: exponential e^z, logarithm log z, complex exponents z^c, c^z, cos z and sin z, know their definitions, properties and derivatives

  11. Logarithms : log z versus Log z, branches and branch cuts of log z, the principal branch of log z, properties of log z

  12. Contour Integrals: Set-up and evaluation of contour integrals, parametrizations of lines and circles, properties of contour integrals, upper bounds for the moduli of contour integrals (the ``ML Theorem'')

  13. Important Integration Theorems and Formulas: AD theorem (antiderivative = path independence, etc.), Cauchy-Goursat theorem, simply versus multiply connected domains, Principle of deformation of paths (PDP), Cauchy integral formula, extension of the Cauchy integral formula

  14. Applications of Cauchy's Integral Formulas: Analytic functions have analytic derivatives of all orders, Liouville's theorem, the fundamental theorem of algebra, the maximum modulus principle

  15. Series: Convergence of, geometric series, Taylor series (how to compute them, know standard examples like e^z), Taylor's theorem, Maclaurin series, Laurent series (how to compute them via substitution, partial fractions or using geometric series), Laurent's theorem, basic facts about power series (e.g., converges to an analytic function inside the circle of convergence), term by term differentiation and integration

  16. Residues, Poles and Singularities: isolated singularities, removable singularities, poles, essential singularities, Picard's theorem, residues (how to compute them using Laurent series, special formula for poles), Cauchy's residue theorem, evaluating improper integrals using residues