Homework 1

Due: April 11, 2014 in class

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1. Page 56, Chapter 2, Section 5, Problem 66

Find x and y as functions of t for the example above, and verify for this case that v and a are correctly given by the method of the example. In the example, z = x + iy = (i+2t)/(t-i), and v = 3/(t²+1) and a = 6/(t² +1)^3/2

2. Page 56, Chapter 2, Section 5, Problem 67

Find v and a if z = (1-it)/(2t+i)

3. Page 57, Chapter 2, Section 6, Problem 1

Prove that an absolutely convergent series of complex numbers converges. This means to prove that the sum of (a_n+ib_n) converges if the sum of
(a_n²+b_n²) ^1/2 converges.

4. Page 57, Chapter 2, Section 6, Problem 5

Test the sum (1/n²+i/n) for convergence

5. Page 66, Chapter 2, Section 10, Problem 11

Find (-8)^1/3

6. Page 67, Chapter 2, Section 10, Problem 29

Show that the center-of-mass of three identical particles situated at z₁, z₂, and z₃ is (z₁+z₂+z₃)/3.

7. Page 77, Chapter 2, Section 16, Problem 1

Show that if the line through the origin and the point z is rotated by 90^o about the origin, it becomes the line through the origin and the point iz. This fact is sometimes expressed by saying that multiplying a complex number by i rotates it through 90^o. Use this idea in the following problem. Let z = ae^iwt be the diplacement of a particle from the origin at time t. Show that the particle travels in a circle of radius a at velocity v = wa and with acceleration of magnitude v²/a directed toward the center of the circle.

8. Page 79, Chapter 2, Section 16, Problem 11

Prove that

cos A + cos 3A + cos 5A + ... + cos (2n-1)A = sin 2nA/(2 sin A)
sin A + sin 3A + sin 5A + ... + sin (2n-1)A = sin²nA/(sin A)


Hint: Use Euler's formula and the geometric progression formula.

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Solutions: Part 1 and Part 2.

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