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# Senior Mathematics Seminar MATH 4513

OU

GPA 3.99

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This 2 page Class Notes was uploaded by Einar Nitzsche on Monday October 26, 2015. The Class Notes belongs to MATH 4513 at University of Oklahoma taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/229298/math-4513-university-of-oklahoma in Mathematics (M) at University of Oklahoma.

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Date Created: 10/26/15

Math 4513 Problems involving Complex Numbers and Functions 92904 Throughout these problems 2 denotes a complex number with rectangular form 2 z yi 1 Give examples showing that sometimes Argzw equals Argz Argw and sometimes it doesn t to What is and argz when 2 7 i6 OJ Determine the polar form of 1 i For each integer n express 1 in polar form Then use your answer to describe the rectangular form for 1 Plot the complex numbers 1 for a few values of n and see if you can describe the general pattern q lf Euler s identity 5 cos0 i sin0 is squared we get cos 20 l i sin 20 5w cos0 isin02 a Expand out the right hand side of this equation and explain how this proves the double angles formulas for cos and sin b Look at 5193 and determine triple angle formulas for sin and cos 0 In the set of real numbers the equation 5 1 has only one solution but in the set of complex numbers 25 1 has more than one solution Write out all of these complex solutions Suggestion Start with 2 written in polar form a Let fz 22 and consider the associated mapping w 22 from the z plane to the w plane a Determine where the positive and negative real axis in the z plane gets mapped in the w plane b Repeat a with the positive and negative imaginary axis c Where does the circle with radius r0 centered at the origin in the z plane get mapped d Express the function fz fz in rectangular coordinates that isiif w 22 and we write w u 212 what do u and 1 equal in terms of z and 34 and use this to determine where the horizontal line y 3 gets mapped 7 Repeat the previous problem with the function 92 2w

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