HW1s - Key to Complex Analysis Homework 1 Chapter 1 (in...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Key to Complex Analysis Homework 1 Chapter 1 (in Greene & Krantzs book) 13. Give all possible polar forms of each number. solution. (a) z = 3 + i : r = | z | = 3 + 1 = 2 = arg( z ) = arctan( = z < z ) + 2 k = 6 + 2 k, k Z Hence, all the possible polar form of z can be written as: z = re i = 2 e 6 i +2 ki , k Z . (d) z = 4- 8 i : r = | z | = 16 + 64 = 4 5 , = arg( z ) = arctan( = z < z ) + 2 k =- arctan(2) + 2 k, k Z Hence, all the possible polar form of z can be written as: z = re i = 4 5 e- arctan(2) i +2 ki , k Z . 2 14. Convert the polar forms into rectangular forms. solution. (b) z = 4 e i/ 4 Since r = 4 , = / 4, we have that x = r cos( ) = 4 * 2 2 = 2 2 y = r sin( ) = 4 * 2 2 = 2 2 Hence, z = 2 2 + 2 2 i. (g) Since r = 7 , =- / 6, we have that x = r cos( ) = 7 * 3 2 = 7 3 2 y = r sin( ) =- 7 * 1 2 =- 7 2 Hence, z = 7 3 2- 7 2 i. 1 2 28. Compute each of the following derivatives: (a) z ( 4 z 2- z 3 ) (b) z ( z 2 + z 2 z 3 ) (c) 3 x 2 y ( 3 z 2 z 4- 2 z 3 z + z 4- z 5 ) (d) 5 z 3 z 2 ( z 2- z z + 4 z- 6 z 2 ) Solution. We use the fact that z z = 1 , z z = 0 , z z = 0 , z z = 1 , as well as the usual sum and product rule for differentiation....
View Full Document

Page1 / 6

HW1s - Key to Complex Analysis Homework 1 Chapter 1 (in...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online