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Student's Solutions Manual s to accompany Complex Variables and Applications Eighth Edition James Ward Brown University of Michigan — Dearborn Ruel V. Churchill Late Professor of Mathematics University of Michigan Prepared by James Ward Brown University of Michigan — Dearborn McGraw-Hill Higher Education Boston Burr Ridge, IL Dubuque, IA NewYork San Francisco St. Louis Bangkok Bogotá Caracas Kuata Lumpur Lisbon London Madrid Mexico City Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto Contents 1 Complex Numbers 1 2 Basic Algebraic Properties ........cemsereseecereereareeseesenercerrammeronsensos 1 3 Further Properties.............rreeresereneeercentereaneaceseencersensaceesenenses 2 4 Vectors and Moduli................i is ereemeeresereererararerenrererearerarencararaos 3 5 Complex Conjugates .............. ser eeeeserereerenerereneererenserenerserensacaros 5 8 Arguments of Products and Quotients..........eesenerseressareeseseensenes 8 10 Examples ..........crerermereesereamereerrareasereacenecareasereencensarsarmenses 12 11 Regions in the Complex Plane... 2 Analytic Functions 22 12 Functions of a Complex Variable..........sieemeeasesseneasarereemes 22 18 Continuity 20 Differentiation Formulas ...................e src ceeeeererreareerentenor 24 23 Polar Coordinates ............... rererreemenenserenereeneesrereencarerseseensensor 25 25 Examples... rrermeeeecermeereecercereesserercencenraeescaerserenaseass 31 26 Harmonic Functions ............ereseeereereerenrerceneeeaceeseneersesennerasos 32 3 Elementary Functions 35 29 The Exponential Function... 31 Branches and Derivatives of Logarithms................... 39 32 Some Identities Involving Logarithms................... eee 41 33 Complex Exponents ............cereereereeerenceeserenecencersereereararas 43 34 Trigonometric Functions .............eeeemeeeeeserenereecererrererearereroo 45 35 Hyperbolic Functions... ir cereseerenserreeeencersecrecreenrersos 49 iii 84 Integration Along a Branch Cut... rererrerrerermeena 138 85 Definite Integrals Involving Sines and Cosines 87 Rouché's Theorem..............sreremerereaceenereerreneensersercaneersaso 153 89 Examples...........ereereeneeneereaereeeeence rear rereeroncensersorenaaaasaesaaso 155 Note to the reader: The numbering system used here to identify chapters, sections, and exercises is consistent with that used in the text. For instance, according to the table of contents just above, solutions of exercises following Section 10 in Chapter 1 of the text start on page 12 of this solutions manual, (b) To verify the distributive law, write Aa + 2) = (0,00 0) + (ap )] = (0, O + 2203 + 35) = (06 + AX — 534 — Pas DM + + 235) = (0 FAX po VMA, + AA) = (0% — 3% 3% + A) +, — 3a, o +) = (1,3) 1) + (2, 9t,, 9) = 774 + Zzo- 9%. CDr=CLOgp)=(-x,—y)=-2. 11. The problem here is to solve the equation 2? +z+1=0 for z=(x,y) by writing (6, 9)00,3) + (1,9) + (1,0) = (0,0). Since 3-4 x+L2x7+9)= (0,0), it follows that X-y+x+1=0 and 2x9+9=0. By writing the second of these equations as (2x+1)y=0, we see that either 2x+1=0 or y=0. If y=0, the first equation becomes x? +x+1=0, which has no real roots (according to the quadratic formula). Hence 2x+1=0, or x=-1/2. In that case, the first equation reveals that )? = 3/4, or y= 3/2. Thus 1,3 =(4,)=|-5£D|. z=(x,)) ( Z 3) SECTION 3 1 (a) l+2i (2-1; (0+2N3+40) (2-5X5i) -5+10 -5-10) 2, 3-4 Si (3-43+4) (5X) 25 25 5 Si 5i Si 1 (b) - - — = A En ES 1-)2-DG-D) U-3DG-) 0 2 (co) G-Dt=[4-Dd-DP=(2))=—4. 1 pot l/z É [= (220). PARES 3. (uai) = aluluzl= az) ]= auz)z)]= alza(g2)]= (az). EA grlg)s como FADIRA FA EA Z3 Au Tl) UU ODDS arm wa lo Az mw) Az % FA Ea SECTION 4 2 1. (a) q=2i, g=q-i y Ut] % Ut 0 x Z> (b) u=(V31, w=(43,0) » Ed] uta 5. (a) Rewrite lz-I+il=laslz=(1-)=1. This is the circle centered at 1—i with radius 1. T is shown below. o ” 6. (a) Write lz—4i+z+4il=10 as Iz— 4il+Iz—(—4i)l= 10 to see that this is the locus of all points z such that the sum of the distances from z to 4i and —4i is a constant. Such a curve is an ellipse with foci +4i. (b) Write lz— l=lz+il as iz — 1=Iz = (=i)l to see that this is the locus of all points z such that the distance from z to 1 is always the same as the distance to —i. The curve is, then, the perpendicular bisector of the line segment from 1 to —i. SECTION 5 (c) Q+iP=(2ri) =(2-) =4 dit =4-4i-1=3-4i; (d) [27+5JN2 — Dl=I27+ 5112 —il=lZz+5IV241 =3 127+5]. 2. (a) Rewrite Re(z-i)=2 as Rejx+i(-y—-D]=2, or x=2. This is the vertical line through the point z=2, shown below. (b) Rewrite [27 +il=4 as 2]z +iF 4, or radius 2, shown below, 2 Pp 7 Write z =x, +iy, and z,=x,+iy,. Then “= (n ti) += (x) +04) =(6-3)-Oy)=0—)-(r-i)=7—% and Tião = Oh HO + Do) = (ão — 3135) +IQHA, +) = (0% — o) = On, + 5) = (4 — Da, — yo) = 2,5. (a) um =(o)a=um%= (x FAIA =&%% Za 2225 (b) F=rr=r =uu=(za(za=ama2=0". vu) Um Um (b) al tdo dal Z%23) xl Izollz;) In this problem, we shall use the inequalities (see Sec. 4) IRezislzl and ia +% +43 -isin i —2i sin£ 2 or .0.. (Qn+Do], [e (2n+1)9 sin +sin +1 Cos, = 0857 2 2sin Ê 2 The real part of this is clearly psin Qn+16 =+ ——, 2 2sin— 2 and we arrive at Lagrange's trigonometric identity: sin Que DO 1+0059+005264---+cosn6 = 14 —— 2 — (0<0<27). 2sin — 2 We know from de Moivre's formula that (cos 8 + isin 0) = cos30 + isin39, or cos? 8+3cos* B(isin 0) + 3cos Olisin 9) + (isin 6) = c0s30 + isin30. That is, (cos” 8 — 3cos Osin? 9) + i(3cos” Bsin 6 — sin? 8) = cos36 + isin30. By equating real parts and then imaginary parts here, we arrive at the desired trigonometric identities: (a) cos30 = cos" B —3cosBsin? 8; (b) sin30=3cos” OsinB—sin” 6. 1 SECTION 10 1. (a) (b) Since 2i = zenfí(Z + 2ta)] (k =0,+1,t2,...), the desired roots are Qu” = Benl(S + +) That is, co=2e"! = V(cosE risinZ)= “(7+) =1+4i and q = (42 = e, =, co being the principal root. These are sketched below. Observe that 1-3; = 2enfÁ-= + 2ha) (k=0,+1,+2,...). Hence 1-3) =/2 enfi(-Z + tz) The principal root is q=NZe ts = v(cosE - isinã) = and the other root is q = (20 "e" =-c,=—-— These roots are shown below. (k=0,1). (k=0,1). 14 The others are q=Qe "er? =ci=1+3i, E = (Set = co =43 1), 6, = (Lee = c(i)=—(1+3i). These roots are all shown below. a E 3. (a) By writing —1=lexpli(x+2km)] (k=0,+L+2,...), we see that corcel] The principal root is ; To Tm 1+V3i =e"P=cos>+isino= 3 . 3 3 2 The other two roots are q=ef=- and , or Fra om 1 c = e5t3 =ee imtã =cos>-isin—= ? 3 3.2 All three roots are shown below. (k=0,1,2). 15 (b) Since 8 = 8expli(O + 2km)] (k = 0,+1,+2,...), the desired roots of 8 are gs =2 exofi =) (k=0,1,2,3,4,5), the principal one being =". The others are e NZet = va(cos5 +isinã) = (202) = e , Cc =(N2e Te = va(cosã -isin kn = (2-1) = a , a= 26" =-V2, cs = (ND ye ng, = Lt E and : i 1-3 c;=(N2e yet = —c, = E" All six roots are shown below. a The three cube roots of the number z, = -4/2+4N2i= Besp(i =) are evidently (9) = 2em(5 +28) (k=0,1,2). In particular, = zex(12) = v2a +), 
