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Unformatted text preview: Complex Exercises 1. Show where in the complex plane are: 1, i, -1 + 3i, i , i , and write all these numbers in the form rei q . 2. State the rule for mult iplying two complex numbers of the form rei q , and fro m that figure out the inverse of a complex number: that is, express 1 / ( rei q ) as r1 ei q . 1 3. Find how to invert a number in the other notation: if Hint: it helps to mult iply 1 = a + ib find a, b in terms of x, y. ,
x + iy 1 x - iy by . x + iy
x - iy 4. Show on a diagram where in the co mplex plane is a cube root of 1, we’ll call it w How many .
cube roots does 1 have? Show all possibilit ies on the diagram. Next, what about cube roots of 1? Show them on another figure. (Note: w is co mmonly used for a cube root of 1. we also use it, of course, for angular frequency. Take care not to confuse the two.) 5. Draw a complex number z as a vector (point ing fro m the origin to z), then draw on the same diagram as vectors iz, z/i, w z. ( w being the cube root of 1.) 6. Using eiq = cos q + i sin q , from ei( A+ B ) = eiA eiB , deduce the formulas for
sin ( A + B ) , cos ( A + B ) . 7. Suppose the point z moves in the co mplex plane is such a way that at time t z ( t ) = Aeiw t , and 0 A is real and w0 = 2p sec 1 . Where is z at t = 0? Where at t = 1 second? Where at t = 0.5 seconds? Where at t = 0.25 seconds? Describe how z moves as time progresses. How would your answer change if A were pure imaginary instead of real? 8. Consider again z ( t ) = Aeiw t , w0 = 2p sec 1 . Differentiate both sides to find an expressio n for 0 &
the velocit y z ( t ) = dz / dt as the point moves alo ng its path. How does the velocit y vector relate to the posit ion vector? Next, find by differentiating again the acceleration vector, and comment on its direct ion. 9. State briefly how z behaves in time if z ( t ) = Aeiw t for real w . How would this behavior change if w had a small imaginary part, w = w0 + iG , where G is small? Sketch how z would mo ve in the co mplex plane, both for G posit ive and G negat ive. 10. Consider the quadratic equat ion x 2 - 2bx + 1 = 0 . For b = 1, both roots equal 1. Sketch (in the complex plane) how the larger root moves as b varies fro m 1.2 down through 1 to 0.8. When you’ve done that, do the same for the other root, preferably in a different color. ...
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- Fall '07