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Math 412 Final Exam Information Spring 2009
WHEN: Friday, May 22 from 9:30-11:30 a.m.
WHERE: Usual classroom: MH-
COVERAGE: The final exam is cumulative. It will cover everything we have discussed during lecture throughout the semester. Topics in the book that we did not cover in class will not occur on the final–consult your class notes for this. The more time and energy we spent on a topic, the more likely it is to appear on the final exam. There may be a slight emphasis placed on material covered since the second midterm (i.e. Sections 6.6, 7.2-7.4, and 8.1). Please consult the two midterm review packets to see which sections were omitted for the midterms–those sections will also be omitted on the final exam. When in doubt, consult the in-class notes or ask me.
EXTRA OFFICE HOURS:
Saturday, May 16, 11-1 p.m.
Sunday, May 17, 12-3 p.m.
Monday, May 18, 10-2 p.m.
Tuesday, May 19, 10-12 noon and 7-9 p.m.
Wednesday, May 20, 10-2 p.m.
Thursday, May 21, most of the day after I arrive
REVIEW SESSION:
Saturday, May 16, 1-3 p.m. (MH-416)
THEOREMS FOR THE FINAL EXAM: Recall that I will choose four of the six theorems below to put on the final:
- Theorem 3.
- Theorem 5.
- Theorem 6.
- Theorem 6.
- Theorem 6.
- Theorem 6.
Note: The statements of the four chosen theorems WILL appear on the final exam itself. You will be asked to prove your choice of two of them. They will be graded fairly closely, so please pay attention to detail. Any questions about the proofs? Ask me!
STUDYING: All comments made on the review packets for Midterm 1 and Midterm 2 still apply– please review! I recommend going over the midterms and sample midterms again, along with as many quizzes, practice quizzes, and groupworks that you have time for and that you think you need.
Here is an overview of the topics we have covered since the second midterm (go back to the midterm review packets for the older topics). You should be comfortable with all of the following words below:
Chapter 6: Morera’s Theorem (Theorem 6.13), Gauss’ Mean Value Theorem (Theorem 6.14), Max- imum Modulus Principle (Theorems 6.15 and 6.16), Cauchy’s Inequalities (Theorem 6.17), Liouville’s Theorem (Theorem 6.18), Fundamental Theorem of Algebra (Theorem 6.19)
Chapter 7: Taylor series for f centered at α, Maclaurin series for f , Taylor’s Theorem (Theorem 7.4), Radius of convergence of the Taylor series for f centered at α (Corollary 7.3), Laurent series, Annulus, Laurent’s Theorem (Theorem 7.8), Singularity: isolated, removeable, essential, pole of order k.
Chapter 8: Residue of f at α: Res[f, α], Cauchy’s Residue Theorem (Theorem 8.1), Formula for Res[f, α], where α is a pole of order k (Theorem 8.2).
THINGS TO BE ABLE TO DO: (Again, consult the midterm review packets for help with this too–the list below only pertains to topics since the second midterm.)
- Factor polynomials into a product of linear factors, as per the Fundamental Theorem of Alge- bra.
- Apply the Maximum Modulus Principle to help locate where an analytic function assumes its maximum value on a closed region R.
- Use the Maximum Modulus Principle to argue that a function must be constant, or must be unbounded, or whatever....
- Use Cauchy’s Inequalities to place a bound on the modulus of higher derivatives of an analytic function f.
- Compute Taylor and Maclaurin series centered at given α ∈ C for given functions f (z), and state their domain (i.e. disk) of convergence.
- Use the formula for the coefficients of a Taylor series to compute f (n)(α) for given n ∈ N.
- Find Laurent series expansions for f (z) around a point α and state their domain (i.e. annulus) of convergence.
- Be able to locate and classify singularities of a function f (z).
- Be able to compute the residue of a given function f (z) at a given point α ∈ C (e.g. Theorem 8.2).
- Be able to use Cauchy’s Residue Theorem to compute contour integrals (e.g. Theorem 8.1).
ADVICE:
Important Suggestion: Try to re-do problems from scratch, rather than just looking up solutions that I’ve posted. General rule of thumb: The more time we spent on a topic or type of problem, the more likely it is to appear in some form on the exam.
