HW4 - 2+2 n ) i . = log( i 1 / 2 ) = (1 / 4 + n ) i . And 1...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
MATH 185 SOLUTIONS OF HW IV (1) (24. 7) (a) f ( z ) is real-valued in D . = f ( z ) = f ( z ). By the main result in Ex 3, Sec 24, f is constant in D since f and f = f are both analytic in D . (b) If c = 0 then | f ( z ) | = 0, i.e. f ( z ) 0 in D . Hence f is constant. Otherwise f ( z ) = c 2 /f ( z ) is well-defined and analytic. By the main result in Ex 3, Sec 24, f is constant in D (2) (33.13) By using the formula (11),(12) on page 102, we can easily show that sin(¯ z ) = sin z and cos(¯ z ) = cos z . If sin(¯ z ) is analytic at some point p then there is an open neighborhood D of p such that sin(¯ z ) is analytic on D , i.e. sin z is analytic on D . By the main result in Ex 3, Sec 24, sin z is constant on D . But sin z is not constant in any open region, so this is contradiction, i.e. sin(¯ z ) is not analytic at any point. Similarly, cos(¯ z ) is not analytic. (3) (30.5) (a) i 1 / 2 = e 1 / 2 log i = e 1 / 2(1 /
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2+2 n ) i . = log( i 1 / 2 ) = (1 / 4 + n ) i . And 1 / 2 log i = 1 / 2(1 / 2 + 2 n ) i = (1 / 4 + n ) i . Hence log( i 1 / 2 ) = 1 / 2 log i . (b) log( i 2 ) = log(-1) = (-1 + 2 n ) i . But 2 log i = 2(1 / 2 + 2 n ) i = (1 + 4 n ) i . Hence log( i 2 ) 6 = 2 log i . (4) (32.2) (a) i i = e iLogi = e i ( i/ 2) = e-/ 2 . (b) e 2 (-1- 3 i ) = e e (-2 i/ 3) . = [ e 2 (-1- 3 i )] 3 i = e 3 iLog [ e 2 (-1- 3 i )] = e 3 i (1-2 i/ 3) = e 3 i e 2 2 =-e 2 2 since e 3 i =-1. (c) (1-i ) 4 i = e 4 iLog (1-i ) = e 4 i (ln 2-i/ 4) = e +2 ln 2 = e (cos(2 ln 2) + i sin(2 ln 2)) since 4 ln 2 = 2 ln 2. Date : today. 1...
View Full Document

This note was uploaded on 09/01/2010 for the course MATH 185 taught by Professor Lim during the Fall '07 term at University of California, Berkeley.

Ask a homework question - tutors are online