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MAT334 Final Exam Review Sheet

0 Disclaimer

Any of the computation tricks you have learned in first or second year calculus are fair game. You should know how to integrate and differentiate. I will not ask anything really obscure, but if I’ve mentioned it in lecture or asked homework questions about it, then it’s not obscure. This is a calculus course, and you need to be able to do single variable calculus to succeed.

        I have attempted to make this a comprehensive list of topics. I may have missed something. I do not promise that this is exhaustive.

        A note about homework: You may use results from homework, unless I am asking you to reprove them. The homework problems have touched on basically every topic we’ve seen in the course. Reviewing them might be helpful in your studying. (Also, reviewing the feedback you’ve been given on your homework might be helpful too. That could indicate topics you’re shaky on.)

1 Definitions and Theorems

I will be asking you to state definitions and theorems on the test. Know them. Anything is fair game.

2 Complex Algebra

You will need to be able to perform complex algebra fluently, including knowing when working in rectangular form is appropriate vs. knowing when working in polar form is appropriate. You will need to be able to find nth roots of a complex number, and solve simple polynomial equations, including using the quadratic formula. You should know how to work with the modulus |z| and the real and imaginary parts of z, and how they relate. You should be able to prove simple facts about complex numbers.

        You should know the two different ways of writing polar form, and when it is appropriate to use each.

3 Functions and Limits

You should know to find the domain of a function, the range of a function. You should be familiar with e^z and all of the functions based on it: sin z, cos z, sinh z, cosh z, etc.

        You should be able to take limits, including all the tricks from 135. When you think a limit doesn’t exist, you should have some idea of which directions to approach from to get different values for the limit.

4 Logarithms and Branches

You should know what log z = w means. You should know why log is not a function on C and be able to talk about taking a branch of the logarithm. Be able to find log0 z for any branch of the logarithm.

        You should know what a^z means, and why it has multiple branches. Know how it is related to log a. Be able to find all the values of the expression a^z.

        You should be able to find the branch cuts of functions constructed out of logarithms.

5 Differentiation and Analyticity

You should know the limit definition of a derivative. You should know the Cauchy-Riemann equations and what information they give you. You should know what it means to be analytic on a domain, and how we check for that using Cauchy-Riemann and continuity. You should know that we never use the limit definition when Cauchy-Riemann suffices. Know what entire functions are, and which of our common functions are entire.

        You should know all of the common derivatives: polynomials, rational functions, power z^a and expo-nentials a^z, logarithms, and trig functions. You should know which differentiation rules are valid, and what they say.

        You should know what it means for a real function of two variables to be harmonic, and be able to find harmonic conjugates. You should know when two harmonic functions u, v give an analytic f = u + iv.

        You should know that analytic functions are smooth and what this says about their real and imaginary parts.

        You should know Liouville’s theorem and what it says about entire functions. You should know how to use it to tell if an entire function is bounded. This theorem is particularly useful in cooking up good proof questions. You should know the Fundamental Theorem of Algebra.

6 Integration

You should be familiar with a variety of techniques for integration: definition, the Cauchy integral theorem, the Cauchy Integral formula, the general Cauchy integral formula, and the residue theorem.

        To that end, you should be able to recognize when a function is analytic on a domain, be able to pick out discontinuities that prevent you from using the CIT. You will need to be able to use the deformation theorem to handle having multiple discontinuities inside your curve, as well as knowing how to reduce an unpleasant curve into a manageable one.

        You should know how integrating over a negatively oriented curve affects the integral, as well as how to handle curves that are not simple.

        You should know how to justify your use of theorems to calculate integrals. I will expect you to verify the hypotheses of theorems before using the theorems.

        You should know how to use the residue theorem to compute integrals. As such, you should know how to calculate residues. Know the formulas for removable discontinuities and poles. Know how to use Laurent series to handle essential singularities.

7 Power Series

You should be able to use various tests to see whether a series converges or not, as well as to compute the values of some convergent series. You will need to know the divergence test, the ratio test, and about geometric series.

        You should know how to find the radius of convergence of a power series, how to find a power series for f(z) at an arbitrary center and you should know the basic easy examples: e^z, sin(z), cos(z), , any polynomial. You should be able to take derivatives of and find primitives for power series, and how to combine these with series we know to find new power series.

8 Laurent Series and Residues

You should now how to find a Laurent series expansion for types of functions we’ve encountered. Specifically, know how to use the expansion for  on |z| > 1, as well as how to work with functions like , where you use a power series you already know. You should also know the explicit formula for a−n. Especially, you should know how a−1 can be used to compute integrals.

        You should know how to figure out if a point is an isolated singularity of a function, and how to distinguish what type.

        You should know how to compute residues around all types of singularities: removable, poles, and essential singularities. Know when Res(f; z0) is equal to a−1 in a Laurent series for f(z) centered at z0.

9 Contour Integrals

You should know how to compute improper integrals  r(x)dx where r(x) is a rational function, using complex integration. You will receive ZERO marks for using other methods for computing such integrals.

        You should know how to bound r(z) on |z| = R using the triangle inequality. You should know the correct contour to integrate over, and how it relates to the integral you’re actually interested in. You should know how to show the improper integral exists, if necessary.

        You should also know how to handle contour integrals involving sin(x) and cos(x), and the trick involving e^ix to work with them. Know why this trick is necessary.