2 show that if r is any number such that a 1 r a 2 r

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2. Show that if R is any number such that | a 1 | R + | a 2 | R 2 + · · · + | a n | R n < 1 , then all the roots of z n + a 1 z n - 1 + · · · + a n = 0 are in absolute value less than R . In particular show that all the roots of z 5 - 13 z - 7 = 0 are in absolute value less than 2 1 67 . 3. Determine the numbers of the roots of the equation z 2 p + az + b = 0 where a and b are real and p odd, which have their real parts positive and negative. Show that if a > 0, b > 0 then the numbers are p - 1 and p + 1; if a < 0, b > 0 they are p + 1 and p - 1; and if b < 0 they are p and p . Discuss the particular cases in which a = 0 or b = 0. Verify the results when p = 1. [Trace the variation of am( z 2 p + az + b ) as z describes the contour formed by a large semicircle whose centre is the origin and whose radius is R , and the part of the imaginary axis intercepted by the semicircle.] 4. Consider similarly the equations z 4 q + az + b = 0 , z 4 q - 1 + az + b = 0 , z 4 q +1 + az + b = 0 . 5. Show that if α and β are real then the numbers of the roots of the equation z 2 n + α 2 z 2 n - 1 + β 2 = 0 which have their real parts positive and negative are n - 1 and n + 1, or n and n , according as n is odd or even. ( Math. Trip. 1891.) 6. Show that when z moves along the straight line joining the points z = z 1 , z = z 2 , from a point near z 1 to a point near z 2 , the increment of am 1 z - z 1 + 1 z - z 2
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APPENDIX I 552 is nearly equal to π . 7. A contour enclosing the three points z = z 1 , z = z 2 , z = z 3 is defined by parts of the sides of the triangle formed by z 1 , z 2 , z 3 , and the parts exterior to the triangle of three small circles with their centres at those points. Show that when z describes the contour the increment of am 1 z - z 1 + 1 z - z 2 + 1 z - z 3 is equal to - 2 π . 8. Prove that a closed oval path which surrounds all the roots of a cubic equation f ( z ) = 0 also surrounds those of the derived equation f 0 ( z ) = 0. [Use the equation f 0 ( z ) = f ( z ) 1 z - z 1 + 1 z - z 2 + 1 z - z 3 , where z 1 , z 2 , z 3 are the roots of f ( z ) = 0, and the result of Ex. 7.] 9. Show that the roots of f 0 ( z ) = 0 are the foci of the ellipse which touches the sides of the triangle ( z 1 , z 2 , z 3 ) at their middle points. [For a proof see Ces` aro’s Elementares Lehrbuch der algebraischen Analysis , p. 352.] 10. Extend the result of Ex. 8 to equations of any degree. 11. If f ( z ) and φ ( z ) are two polynomials in z , and γ is a contour which does not pass through any root of f ( z ), and | φ ( z ) | < | f ( z ) | at all points on γ , then the numbers of the roots of the equations f ( z ) = 0 , f ( z ) + φ ( z ) = 0 which lie inside γ are the same. 12. Show that the equations e z = az, e z = az 2 , e z = az 3 , where a > e , have respectively (i) one positive root (ii) one positive and one negative root and (iii) one positive and two complex roots within the circle | z | = 1. ( Math. Trip. 1910.)
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APPENDIX II (To Chapters IX, X) A Note on Double Limit Problems In the course of Chapters IX and X we came on several occasions into contact with problems of a kind which invariably puzzle beginners and are indeed, when treated in their most general forms, problems of great difficulty and of the utmost interest and importance in higher mathematics.
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