{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

118Homework2

# 118Homework2 - Homework 2 Due Wednesday Sept 8 1 Suppose...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Homework 2 Due Wednesday, Sept. 8 1. Suppose that f : C → C and g : C → C and that lim z → z f ( z ) = 0. Prove that if there are positive real numbers M and r such that | g ( z ) | ≤ M whenever | z- z | < r , then lim z → z f ( z ) g ( z ) = 0. (Note that if f ( z ) = z and g ( z ) = 1 /z , then lim z → f ( z ) = 0 but lim z → f ( z ) g ( z ) = 1 6 = 0. Why doesn’t this contradict the result you have proven above?) 2. Define f : C → C by f ( z ) = z 3 . (a) Use the definition of the derivative to calculate f ( z ). (b) Writing f ( z ) = u ( x, y ) + iv ( x, y ) verify that the Cauchy-Riemann equations hold, and that f ( z ) = u x ( x, y ) + iv x ( x, y ) = v y ( x, y )- iu y ( x, y ) . 3. Let g : C → C by g ( z ) = z 2 . Use the Cauchy-Riemann equations to show that for z 6 = 0 the derivative g ( z ) does not exist. Also prove that g is differentiable at 0. (Hint: Use the definition of g (0).) 4. For f : C → C with f = u + iv define the conjugate of f to be the function...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

118Homework2 - Homework 2 Due Wednesday Sept 8 1 Suppose...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online