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# Lee: denote any nonzero complex number. written.: = n/H

ISBN: 9780073383170 169

## Solution for problem 3.36 Chapter Chapter 3

Complex Variables and Applications | 9th Edition

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Problem 3.36

Lee: denote any nonzero complex number. written.: = n/H (-;r < (o-) ~ ;r ). and let /1 denote any tixed positive integer ( /1 = I . 2 . ... ) . Show that all of the values of log(: I.'" ) are given by the equation I' I (-) + 2(/HI + k)JT log(: ' 11 ) = -lnr + i------where p = 0. I. 2 .... and k = 0. I. 2 . .... /1 - I. Then. after writingI I .0>+ 2q~- log : = - In r + 1 ----11 II II\llAP. 3where'/ = 0. I. 2 ..... show that the set of values of log(: 1 in) is the same as the setof values of ( l/ 11) log:. Thus show that log(: 1... ,, ) = (I/ 11) log: where. correspondingto a value of log(: 1 :'II) taken on the left. the appropriate value of log: is to he selectedon the right. and conversely. IThe result in Exercise 5. Sec .. B. is a special case of thisone.IS11ggeJfio11: L:se the fact that the remainder upon dividing m1 integer by a positiveinteger 11 is al\vays an integer between 0 and /1 - I. inclusive: that is. when a positiveinteger 11 is spccilled. any integer l/ can be written c/ = p11 + k. where p is an integerand k has one of the values k = 0. I. 2 . .... /1 - I.

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Lee: denote any nonzero complex number. written.: = n/H