The following pages contain detailed solutions of the “ Focus-on-Concepts” exercises found in the undergraduate book “ A First Course in complex Analysis with Applications ” by Dennis G. ill and !atrick D. "hanahan. #e try to $ake the ideas behind each solution as ob%ious as possible by elaborating on details such that no $issing-links are to be supplied by the reader.
Section 1.1 Focus-on-Concepts &'. #hat can be said about the co$plex nu$ber z if
z = z
( )f z *= z * (
&+. Think of an alternati%e solution to !roble$ *&. Then without doing any significant work, e%aluate / i0'&1& .
&2. For n a nonnegati%e integer, i n can be one of four %alues3 , i, -, and -i. )n each of the following four cases, express the integer exponent n in ter$s of the sy$bol k , where k 4 1, , *, . . . . a0 i n 4 b0 i n 4 i c0 i n 4 - d0 i n 4 -i
&5. There is an alternati%e to the procedure gi%en in 20. For exa$ple, the 6uotient '+ i / i $ust be expressible in the for$ a / ib3 ' + i .i
= a ib
Therefore, ' +i = i a i b . 7se this last result to find the gi%en 6uotient. 7se this $ethod to find the reciprocal of 8 9 &i. "olution is $ere nu$erology.
&:. ;ssu$e for the $o$ent that i $akes sense in the co$plex nu$ber syste$.
'1. "uppose z and z* are co$plex nu$bers. #hat can be said about z or z* if z z* 4 1(
'. "uppose the product z z* of two co$plex nu$bers is a non=ero real constant. "how that z *= k z , where k is a real nu$ber.
'*. #ithout doing any significant work, explain why it follows i$$ediately fro$ *0 and 80 z z z . . z *=* ℜ z. that z . * *
'8. >athe$aticians like to pro%e that certain “things” within a $athe$atical syste$ are uni6ue. For exa$ple, a proof of a proposition such as “The unity in the co$plex nu$ber syste$ is uni6ue” usually starts out with the assu$ption that there exist two different unities, say, and * , and then proceeds to show that this assu$ption leads to so$e contradiction. Gi%e one contradiction if it is assu$ed that two different unities exist.
'&. Follow the procedure outlined in !roble$ '8 to pro%e the proposition “The =ero in the co$plex nu$ber syste$ is uni6ue. ”
''. ; nu$ber syste$ is said to be an ordered syste$ pro%ided it contains a subset P with the following two properties3 First, for any nonzero number x in the system, either x or −x is but not both0 in P. econd, if x and y are numbers in P, then both xy and x ! y are in P. )n the real nu$ber syste$ the set P is the set of positi%e nu$bers. )n the real nu$ber syste$ we say x is greater than y, written x ? y, if and only if x 9 y is in P . Discuss why the co$plex nu$ber syste$ has no such subset P . @
