ans5 - Math 185 Sample Answers to Problem Set#5 Page 120 2...

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Page 120 2. First, we have z ( φ ( y )) = 2exp( i arctan( y/ p 4 - y 2 )) = 2(cosarctan( y/ p 4 - y 2 ) + i sinarctan( y/ p 4 - y 2 )) . Now draw a right triangle with two sides adjacent to the right angle having lengths y and p 4 - y 2 . Then the hypotenuse will have length 2, and the angle opposite the side of length y will be θ = arctan( y/ p 4 - y 2 ). One can then read off the diagram that sin θ = y/ 2 and cos θ = p 4 - y 2 / 2, and therefore z ( φ ( y )) = 2 p 4 - y 2 2 + i y 2 ! = Z ( y ) . Drawing such a diagram assumes that θ 0. But if θ is in the range from - π/ 2 to 0, then y 0 and p 4 - y 2 0, so sin θ and cos θ would still be correct since they would have the right signs. We can also check that φ 0 > 0 because φ 0 ( y ) = d dy ( y 4 - y 2 ) 1 + ( y 4 - y 2 ) 2 = 4 - y 2 - y (1 / 2)(4 - y 2 ) - 1 / 2 ( - 2 y ) 4 - y 2 4 4 - y 2 = 4 - y 2 + y 2 4 p 4 - y 2 = 1 p 4 - y 2 > 0 for all y ( - 2 , 2). 6. a . To show that z ( x ) describes an arc, we need to show that y ( x ) is continuous. This is obvious except possibly at x = 0, and in that case it follows from the Squeeze Theorem. Indeed, if k = 2 or k = 3 then - x k x k sin ± π x ² x k for all x (0 , 1], and lim x 0 + ( - x k ) = lim x 0 + x k = 0; therefore lim x 0 + x k sin ± π x ² = 0; setting k = 3 then gives continuity since y (0) = 0. This arc intersects the real axis if and only if y ( x ) = 0, which happens if and only if x = 0 or sin( π/x ) = 0. The latter occurs if and only if π/x is an integral multiple of
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ans5 - Math 185 Sample Answers to Problem Set#5 Page 120 2...

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