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Review2 - 18.6 Review of the Second Part of the Lecture...

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18.6 Review of the Second Part of the Lecture 18.6.1 Final Exam The final exam of MATH 529 will take place on Tuesday, May 3, 2011, 8:00 – 11:00 AM, in Phillips Hall, Room 332 (the regular lecture hall). Please read the paragraph on “Final Examinations” in the Academic Procedures section of the 2010–2011 Undergraduate Bulletin (http://www.unc.edu/ugradbulletin/procedures1.html), to be informed about what to do in case you are unable to take the exam at the scheduled time. As stated in the course syllabus, you are allowed to use a summary of the lecture notes on 6 (six) pages (US letter format, double-sided) for the final exam. This is meant as 6 sheets of paper, with text on both sides. The exam will cover topics from Chapter 12 (up to and including 12.10) and 13–17 (not 16.4). The problems will focus on the application of the methods treated in this course; I want to see that students are able to recog- nize which method should be used for a given problem, and that they apply it correctly. I will not ask for proofs, and I will try to avoid complicated aux- iliary calculations (such as partial fraction decompositions, finding roots of high order polynomials, or computing high-order derivatives of complicated functions). There will be 9 problems in the final exam, which means 20 minutes to solve each problem. 18.6.2 Complex Numbers and Functions (13) Complex Numbers. Complex Plane We had constructed the field of complex numbers C by considering ordered pairs ( x, y ) of real numbers. A complex number z C may be represented in various ways in the complex plane C R 2 , such as z = x + iy = r (cos ϕ + i sin ϕ ) = re , i 2 = 1 , (1067) where x := Re z denotes the real part of z , y := Im z the imaginary part, r = | z | = zz = radicalbig (Re z ) 2 + (Im z ) 2 the absolute value, and ϕ = arg z the argument of z . It is defined by the equations cos ϕ = x r , sin ϕ = y r , (1068) and therefore multi-valued, because of the periodicity of sin and cos. We typically restrict the argument to either arg z [0 , 2 π ) or Arg z ( π, π ]. 198
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The latter is convenient when working with complex logarithms, because Arg z = Im(Log z ), and the branch cut of Arg coincides with the branch cut of Log. We have arg z = braceleftbigg Arg z, Im z 0 2 π + Arg z, Im z < 0 , z negationslash = 0 . (1069) The complex conjugate z C of a complex number z C is defined by z = Re z i Im z ; geometrically, this corresponds to a mirroring on the real axis in the complex plane, and therefore z = z Im z = 0. The absolute value | z | = zz induces a metric on C , d : C × C C , d ( z 1 , z 2 ) := | z 1 z 2 | . Thus we can measure distances between complex numbers (Problem Set 7). In particular, we have the triangle inequality | z 1 + z 2 | ≤ | z 1 | + | z 2 | , z 1 , z 2 C . Polar Form of Complex Numbers. Powers and Roots. The polar form of a complex number is useful to represent products, quotients, powers and roots of complex numbers: z 1 z 2 = | z 1 || z 2 | exp ( i (arg z 1 + arg z 2 )) , (1070) z 1 z 2 = | z 1 | | z 2 | exp ( i (arg z 1 arg z 2 )) , (1071) z n = | z | n exp ( in arg z ) , (1072) n z = n radicalbig | z | exp parenleftBig i arg z n parenrightBig exp parenleftbigg i 2 πk n parenrightbigg , k = 0 , . . . , n 1 . (1073) Derivative. Holomorphic Function.
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