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

This preview shows page 43 - 47 out of 102 pages.

We have textbook solutions for you!
The document you are viewing contains questions related to this textbook.
Calculus
The document you are viewing contains questions related to this textbook.
Chapter 4 / Exercise 2
Calculus
Stewart
Expert Verified
| 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 .
We have textbook solutions for you!
The document you are viewing contains questions related to this textbook.
Calculus
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.

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture