{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# hw1_soln - Complex Analysis Fall 2007 Homework 1 Solutions...

This preview shows pages 1–3. 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: Complex Analysis Fall 2007 Homework 1: Solutions 1.1.2. (a) (2 + 3 i )(4 + i ) = (8- 3) + (12 + 2) i = 5 + 14 i (b) (8 + 6 i ) 2 = (64- 36) + (48 + 48) i = 28 + 96 i (c) 1 + 3 1 + i 2 = 1 + 3(1- i ) (1 + i )(1- i ) 2 = 1 + 3- 3 i 2 2 = 5 2- 3 2 i 2 = 25 4- 9 4 +- 15 4- 15 4 i = 4- 15 2 i 1.1.6. (a) If z = x + iy we have z + 1 2 z- 5 = ( z + 1)(2¯ z- 5) (2 z- 5)(2¯ z- 5) = 2 z ¯ z- 5 z + 2¯ z- 5 4 z ¯ z- 10( z + ¯ z ) + 25 = 2 | z | 2- 5 z + 2¯ z- 5 4 | z | 2- 10( z + ¯ z ) + 25 = 2( x 2 + y 2 )- 5 x + 2 x- 5 +- 5 yi- 2 yi 4( x 2 + y 2 )- 20 x + 25 = 2( x 2 + y 2 )- 3 x- 5 4( x 2 + y 2 )- 20 x + 25 +- 7 y 4( x 2 + y 2 )- 20 x + 25 i so that Re z + 1 2 z- 5 = 2( x 2 + y 2 )- 3 x- 5 4( x 2 + y 2 )- 20 x + 25 , Im z + 1 2 z- 5 =- 7 y 4( x 2 + y 2 )- 20 x + 25 . 1 (b) If z = x + iy then z 3 = ( x + iy ) 3 = x 3 + 3 x 2 yi + 3 xy 2 i 2 + y 3 i 3 = ( x 3- 3 xy 2 ) + (3 x 2 y- y 3 ) i so that Re z 3 = x 3- 3 xy 2 , Im z 3 = 3 x 2 y- y 3 . 1.1.18. (a) (1- i )- 1 = 1+ i (1- i )(1+ i ) = 1+ i 2 = 1 2 + 1 2 i (b) 1+ i 1- i = (1 + i )(1- i )- 1 = (1 + i ) ( 1 2 + 1 2 i ) = ( 1 2- 1 2 ) + ( 1 2- 1 2 ) i = i 1.2.2. (a) The equation z 6 + 8 = 0 is equivalent to z 6 =- 8. Since | - 8 | = 8 and arg(- 8) = π , the solutions to the latter equation are z k = 6 √ 8 cos π 6 + πk 3 + i sin π 6 + πk 3 for k = 0 , 1 , . . . , 5. (b) The equation z 3- 4 = 0 is equivalent to z 3 = 4 which, since | 4 | = 4 and arg(4) = 0, has the solutions z k = 3 √ 4 cos 2 πk 3 + i sin 2 πk 3 for k = 0 , 1 , 2. 1.2.4. Recalling that conjugation preserves the arithmetic of C , we have (8- 2 i ) 10 (4 + 6 i ) 5 = (8 + 2 i ) 10 (4- 6 i ) 5 . 1.2.6. DeMoivre’s formula and the binomial theorem give cos 6 x + i sin 6 x = (cos x + i sin x ) 6 = (cos 6 x- 15 cos 4 x sin 2 x + 15 cos 2 x sin 4 x- sin 6 x ) + i (6 cos 5 x sin x- 20 cos 3 x sin 3 x + 6 cos x sin 5 x ) ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 6

hw1_soln - Complex Analysis Fall 2007 Homework 1 Solutions...

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

View Full Document
Ask a homework question - tutors are online