# Z 1 3 i 2 r 1 r 1 r 1 2 we obtain e iz z 2 z 1 1 r 1

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Chapter 4 / Exercise 2
Calculus
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| z |- - 1 - 3 i 2 ! = ( R - 1 )( R - 1 ) = ( R - 1 ) 2 , we obtain e iz z 2 + z + 1 1 ( R - 1 ) 2 for z C R . 10. Let f ( z ) = ( z 2 / | z | if z 6 = 0 0 if z = 0 Show that f ( z ) is continuous everywhere but nowhere analytic on C .
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The document you are viewing contains questions related to this textbook. The document you are viewing contains questions related to this textbook.
Chapter 4 / Exercise 2
Calculus
Stewart Expert Verified
2.3 Analytic functions 43 Proof. Since both z and | z | are continuous on C , z 2 / | z | is continuous on C * . There- fore, f ( z ) is continuous on C * . To see that it is continuous at 0 , we just have to show that lim z 0 f ( z ) = lim z 0 z 2 | z | = f ( 0 ) = 0 . This follows from lim z 0 z 2 | z | = lim z 0 | z | 2 | z | = lim z 0 | z | = 0 . Therefore, f ( z ) is continuous everywhere on C . To show that f ( z ) is nowhere analytic, it suffices to show that the Cauchy-Riemann equations fail for f ( z ) on C * . This follows from x + i y z 2 | z | = 2 z | z | - x z 2 | z | 3 + i - 2 i z | z | - y z 2 | z | 3 = ( 4 z - x - iy ) z 2 | z | 3 = 3 z | z | 6 = 0 for z 6 = 0. Consequently, f ( z ) is nowhere analytic. 11. Find where tan - 1 ( z ) = i 2 Log i + z i - z is analytic?
44 Chapter 2. Functions 12. Show that the following functions are defined on all of C , but are nowhere analytic (here z = x + iy ): (a) z 7→ 2 xy + i ( x 2 + y 2 ) ; (b) z 7→ e y e ix .
13. Show where the function z 7→ x 3 + i ( 1 - y ) 3 is: (b) differentiable (here z = x + iy ).
2.3 Analytic functions 45 which is only satisfied at x = 0 , y = 1; and, on the other hand, u y = - v x 0 = 0 , which holds everywhere. Note also that the componentes of f ( z ) , and all its first- order partials exist everywhere. Since the Cauchy-Riemann equations only hold at z = i , the function f ( z ) is only dif- ferentiable at z = i . Hence, in particular, it is not differentiable on any neighbourhood of any point, and therefore is nowhere analytic.
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