高等数学第四册课后习题答案(1-6章)

1、? 1.(1) 2i; (2) 2 5; (3) i 2; (4) 4; (5)?(x + iy)2= a + bi,? x2 y2= a,2xy = b.(1.1) ?(x2+ y2)2= (x2 y2)2+ 4x2y2= a2+ b2,?x2+ y2= a2 + b2.?(1.1)? ? x2= 1 2(a + a2 + b2), y2= 1 2(a + a2 + b2). ?x?,?y?,?,?(1.1)?.? ?b 6= 0?, a + bi = s a + a2 + b2 2 + i b |b| s a + a2 + b2 2 . ?b = 0?,?a 0?,?a;?a 0?,?ia

2、. 2.(1) 12i 34i 2i 5i = 16+8i 25 ,? 16 25, ? 8 25, ? 85 25 ,?arctan 1 2 + 2k, k = 0,1,2,. (2)? ? 1+3i 2 ?n = ? ei 3 ?n = ei n 3= cos n 3 + isin n 3 , n = 2,3,4?,?1.? ?, 1 + 3i 2 !n = 1 2 + i 3 2 ,n = 2; 1,n = 3; 1 2 i 3 2 ,n = 4. ?n = 2?,?1 2, ? 3 2 ,? 2 3 + 2k, k = 0,1,2,; n = 3?,? 1,?0,?(2k + 1),

3、k = 0,1,2,; n = 4?,?1 2, ? 3 2 ,? 4 3 + 2k, k = 0,1,2,. (3)?1?(5)? 1 + i = s 1 + 2 2 + i s 1 + 2 2 . ?,?1 + i = 2ei 4, ? 1 + i = 4 2ei/4+2n 2, n = 0,1. ?n = 0?,? q 1+2 2 , ? q 1+2 2 ,? 4 2, ? 8 + 2k, k = 0,1,2,; n = 1?,? q 1+2 2 ,? ? q 1+2 2 ,? 4 2, ? 9 8 + 2k, k = 0,1,2,. (4)? 3 + i = 2ei 6, ?(3 +

4、i)3= 23ei 2= i 8. ?(3 + i)3?0?1 8, ? ? 1 8, ? 2 + 2k, k = 0,1,2,. (5)? 1 2, ? 3 2 ,?1,? 3 + 2k, k = 0,1,2,. 3.?z1= ei 4,z2= 2ei 6, ?z1z2= 2ei 12= 2(cos 12 + isin 12), z1 z2 = 1 2e i5 12= 1 2(cos 5 12 + isin 5 12). 4.?z+ 1 z = 2cos?, ? z + 1 z ?2 = z2+2+ 1 z2 = 4cos2.?4? ? z 1 z ?2 = 4sin2. ?z 1 z =

5、2isin.?z = ei.? zm+ 1 zm = eim+ eim= 2cosm. 1 2? 5.(1)?z = 2 1 + 3i= 1 1 2 + 3 2 i = ei ei 3 = ei 2 3 ?argz = 2 3 . (2)? 3 i = 2ei 6, ?z = (3 i)6= 26ei= 26ei,?argz = . 6.(1) i(1 3i)(3 + i) = ei 2 2ei 3 2ei 6= 4ei 3= 2 + 23i. (2) 5i 2 + i = 5ei/2 5eiarctan 1 2 = 5ei(/2arctan 1 2)= 5 ? cos( 2 arctan 1

6、 2) + isin( 2 arctan 1 2) ? = 5 ? sin(arctan 1 2) + icos(arctan 1 2) ? = 5 ? 1 5+ i 2 5 ? = 1 + 2i. (3) (1 + i)7= 27ei73 4= 27ei5 4= 27(2 2 i 2 2 ) = 8(1 + i). (4) (1 + 3i)10 = 210ei 10 3 = 210ei 2 3= 211(1 + 3i). 7.?z4= a4= a4ei?4? zn= aei (2n+1) 4 , n = 0,1,2,3. z0= 2 2 a(1 + i),z1= 2 2 a(1 + i),z

7、2= 2 2 a(1 i),z3= 2 2 a(1 i). 8.(1)?z1= x1+ iy1,z2= x2+ iy2,?Re(z1+ z2) = x1+ x2= Re(z1) + Re(z2).?z1= i,z2= 1 + i, ?Re(z1z2) = 1 6= 0 = Re(z1)Re(z2).?Re(z1z2) 6= Re(z1)Re(z2)?. (2)?. (3) |z1z2| = |z1|z2|?6?.?z1= 1,z2= i,?|z1+ z2| 6= |z1| + |z2|. 9.(4) z1z2+ z1z2= z1z2+ z1z2= 2Re(z1z2) = 2Re(z1z2) =

8、 2Re(z1z2). (6) |z1z2+ z1z2| = 2Re(z1z2) 2|z1z2| = 2|z1|z2| = 2|z1|z2| = 2|z1z2|. (7)?|z1+ z2|2= (z1+ z2)(z1+ z2) = |z1|2+ |z2|2+ z1z2+ z1z2= |z1|2+ |z2|2+ 2Re(z1z2)? 2Re(z1z2) 2|z1|z2|,? |z1+ z2|2 |z1|2+ |z2|2 2|z1|z2| = (|z1| |z2|)2, |z1+ z2|2 |z1|2+ |z2|2+ 2|z1|z2| = (|z1| + |z2|)2. 11.?|z1| = |z

9、2| = |z3| = 1,?z1,z2,z3? ?|z| = 1,?z1= 1.? z1+ z2+ z3= 0?, z2?,?z2= ei(0 ).? z3= (z1+z2)?, z3= ei(+/2).?z2+z3= z1= 1? ?, 1 2( + + 2) = . ?z2? = 2 3 ,?z3? +/2 = 4 3 . ?z1,z2,z3? ?|z| = 1?. 12.(1)?z1?z2?,?,?. (2)?z = 2?,?. (3)?,?. (4)?1.1?,?Rez = 2,Rez = 3?,?. (5)? ?,?,? ?. (6)?Imz = y1?Imz = y2?,?. (

10、7)?1.2?,?. ?3 - O x23 6 y ? ? ? ? ? ? 1 (2,2) ?1.1 - Ox 6 y ?z?z0?, argz?.?argz?,?. 16.?.?|xn x0| |zn z0|?|yn y0| |zn z0|?,?|zn z0| 0?, |xnx0| 0,|yny0| 0.?xn(n = 1,2,)?yn(n = 1,2,)?x0?y0?. ?.?|zn z0| |xn x0| + |yn y0|?,?|xn x0| 0,|yn y0| 0?, |zn z0| 0.?zn(n = 1,2,)?z0?. 17.?z1,z2,z3?,?1.4.?z1z2?z1z3

11、?Arg(z3 z1) Arg(z2 z1) = Argz3 z1 z2 z1 ?(0,)?.?,? Argz3 z1 z2 z1 + Argz1 z2 z3 z2 + Argz2 z3 z1 z3 = Argz3 z1 z2 z1 z1 z2 z3 z2 z2 z3 z1 z3 = Arg(1) = (2n + 1),n = 0,1,2, . ?,?.?. z1? ? ? ? ? z2 z3 ?1.4 ? ?,?,?z = x + iy,f(z) = u(x,y) + iv(x,y). 1.(1)?|z|?.?,?z?,? |z|/z?1?1.?z 0?,?|z|/z?,?|z|? ?.?z

12、?.?z?,?z = ix,? |z + z| |z| z = p(x + x)2 + y2 px2 + y2 x = x + 2x p(x + x)2 + y2+ px2 + y2 x px2 + y2 . ?z?,?z = iy,? |z + z| |z| z = px2 + (y + y)2 px2 + y2 iy = y + 2y i ?p x2+ (y + y)2+ px2 + y2 ? y ipx2+ y2 . ?z 0?,?(|z + z| |z|)/z?,?|z|?.? |z|?z?. ?|z| = px2 + y2,?u = px2 + y2,v = 0.?x2+ y26=

13、0?ux= x px2 + y2 ,uy= y px2 + y2 ,vx= vy= 0.?u,v?C-R?,?|z|? ?.?x = y = 0?u?,?|z|?.?|z|?z ?. ?.?z?.?|z|?z?,?z 0?,? |z + z| |z| z ?.?z?,?,? ?1.?|z|?u?.?u = px2 + y2?,? ?,?.?|z|?z?. (3)?Rez = x,?u = x,v = 0.?ux= 1 6= 0 = vy.?z?, C-R? ?.?Rez?z?. (4)?1/z?,?.?,? 1 z = z |z|2 = x + iy x2+ y2 ,?u = x x2+ y2

14、 ,v = y x2+ y2 .? ux= y2 x2 (x2+ y2)2 ,uy= 2xy (x2+ y2)2 ,vx= 2xy (x2+ y2)2 ,vy= x2 y2 (x2+ y2)2 . ?u,v?C-R?,?1/z?z?. 2.(1)?u = x3,v = y3? ux= 3x2,uy= 0,vx= 0,vy= 3y2. ?3x2= ux= vy= 3y2?,?C-R?.?, u?v?, ?xy?.?f(z) = x3 y3i?. 1 ? ?. 5 6? (2) f(z) = |z|2= x2+ y2.?u = x2+ y2,v = 0. ? ux= 2x,uy= 2y,vx= 0

15、,vy= 0. ?C-R?.?, u?v?,?xy? ?.?f(z) = |z|2?. 3.? ux(0,0) = lim x0 (x)3 (x)2 0 x = 1,uy(0,0) = lim y0 (y)3 (y)2 0 y = 1, vx(0,0) = lim x0 (x)3 (x)2 0 x = 1,vy(0,0) = lim y0 (y)3 (y)2 0 y = 1, ?f(z)?C-R?.? f(z) f(0) z = x3 y3+ i(x3+ y3) (x2+ y2)(x + iy) = x4 xy3+ x3y + y4+ i(x4+ xy3 x3y + y4) (x2+ y2)2

16、 ?z 0?.?,?z?y = kx?,?k? 1 + k k3+ k4+ i(1 k + k3+ k4) k4 . 4.(1)?f(z)?D?, f(z) = ux+ ivx.?f(z) = 0?, ux= vx= 0.?C-R? ux= vy,uy= vx ?uy= vy= 0.?u?v?D?,?f(z)?. (2) f(z) = u iv.?f(z)?D?,?u?v?C-R? ux= vy,uy= vx. ?f(z)?, u?v?C-R? ux= vy,uy= vx. ?vy= vy,vx= vx,?vx= vy= 0.?ux= uy= 0.?u?v?D?, ?f(z)?. (3)?|f

17、(z)| = r,?u2+ v2= r2.?x?y? ( uux+ vvx= 0, uuy+ vvy= 0. ?C-R? ( uux+ vvx= 0, vux uvx= 0. ? ? ? ? ? ? uv vu ? ? ? ? ? = (u2+ v2). ?u2+ v2= r2= 0,?u = v 0,?f(z) 0;?u2+ v2= r26= 0,?, ?ux= vx= 0.?C-R?uy= vy= 0.?u?v?D?,?f(z) ?. ?7 (4)?Ref(z)?D?,?C-R?vx= vy= 0.?v?D?,? ?f(z)?.2 5.?f(z) = ux+ ivx? ?. (1) ex(

18、xcosy y siny)+iex(y cosy +xsiny) = ex(x+iy)(cosy +isiny) = (x+iy)ex+iy= zez.? ?f(z)g(z)= f(z)g(z) + f(z)g(z), (zez)= (1 + z)ez. (2)?f(z) = ux+ ivx? ?.?sinxchy icosxshy. 6.(1)?z2= x2 y2 2xyi, u = x2 y2,v = 2xy. ? ux= 2x,uy= 2y,vx= 2y,vy= 2x. ?C-R?.?z2?,? ?,?z2?. (2)?ez= exiy= ex(cosy isiny), u = exco

19、sy,v = exsiny. ? ux= excosy,uy= exsiny,vx= exsiny,vy= excosy. ?C-R?,?cosy = siny = 0.?y?,?C-R?.? ?ez?. (3)?sinz = eiz eiz 2i = ey+ix eyix 2i = 1 2(e y ey)sinx + i1 2(e y ey)cosx, u = 1 2(e y ey)sinx,v = 1 2(e y ey)cosx. ? ux= 1 2(e y ey)cosx,uy= 1 2(e y +ey)sinx,vx= 1 2(e y ey)sinx,vy= 1 2(e y +ey)c

20、osx. ? ?,?C-R?,?cosx = sinx = 0.?x?,?C-R ?.?sinz?. 7.?x = rcos,y = rsin. u(r,),v(r,)?(r,)? u(x,y),v(x,y)?(x,y)?3.?f(z)?z?,?C-R? u x = v y , u y = v x (2.1) ? u r = 1 r v , v r = 1 r u .(2.2) ?, u r = u x x r + u y y r = cos u x + sin u y , u = u x x + u y y = rsin u x + rcos u y , v r = v x x r + v

21、y y r = cos v x + sin v y , v = v x x + v y y = rsin v x + rcos v y . 2 ?,?. 3 ?.? ?. 8? ?C-R?(2.1)?C-R?(2.2)?.?f(z)?z ?.? u x = cos u r 1 r sin u = cos u r + sin v r , v x = cos v r 1 r sin v = cos v r sin u r . ? f(z) = u x + i v x = cos u r + sin v r + i ? cos v r sin u r ? = (cos isin)u r + (sin

22、 + icos)v r = ei u r + i(cos isin)v r = ei u r + iei v r = 1 ei ?u r + iv r ? . 8.(1)?C-R?, vy= ux= 2x + y.?v = 2xy + y2/2 + (x),?(x)? ?x?.?v?x?,?C-R?,? vx= 2y + (x) = 2y x = uy. ?(x) = x,?(x) = x2/2 + c,?c?.?f(z) = x2 y2+ xy +i(2xy x2/2+y2/2+c).?f(i) = 1+i,?1+i = 1+i(1/2+c),?c = 1/2. ?f(z) = x2 y2+

23、 xy + i(2xy x2/2 + y2/2 + 1/2). (2)?C-R?, vy= ux= 6xy.?v = 3xy2+ (x),?(x)?x? ?.?v?x?,?C-R?,? vx= 3y2+ (x) = 3x2+ 3y2= uy. ?(x) = 3x2,?(x) = x3+ c,?c?.?f(z) = 3x2y y3+ i(x3+ 3xy2+ c).?f(i) = 1,?1 = 1 + ic,?c = 0.?f(z) = 3x2y y3+ i(x3+ 3xy2). (3)?f(z) = i?f(0) = i.?C-R?, vy= ux= 2y.?v = y2+ (x), ?(x)?

24、x?.?v?x?,?C-R? ?,? vx= (x) = 2(1 x) = uy. ?(x) = 2(1 x),?(x) = 2x x2+ c,?c?.?f(z) = 2(x 1)y + i(2x x2+ y2+ c).?f(0) = i,?i = ic,?c = 1.?f(z) = 2(x 1)y + i(2x x2+ y2 1). 9.?.?u = xy2?. ? ux= y2,uy= 2xy;uxx= 0,uyy= 2x. ?x = 0?,?uxx+ uyy= 0?.?,? ? ?uxx+ uyy= 0.?u = xy2?. 10.?u = 2xy,v = x2 y2. ?ux= 2y,

25、vy= 2y.?C-R?y = 0?,? ?.?.?2xy + i(x2 y2)?,?. ?9 11.?f(z)?f(z).?,?f(z) = z,?f(z) = z,?z?,?z? ?19?1?2?.?, f(z)?D?,?f(z)?. ?. ?f(z) = u(x,y) + iv(x,y),?f(z) = u(x,y) iv(x,y) def =(x,y) + i(x,y). f(z)? ?D? u(x,y),v(x,y)?D? ux(x,y) = vy(x,y), uy(x,y) = vx(x,y).(2.3) ?(x,y) = u(x,y),(x,y) = v(x,y),? x = u

26、x(x,y), y = uy(x,y), x = vx(x,y), y = vy(x,y). ?(2.3)? x = y, y = x. ?f(z)?f(z)?. ?g(z) = f(z),? lim zz0 g(z) g(z0) z z0 = lim zz0 f(z) f(z0) z z0 = lim zz0 ?f(z) f(z 0) z z0 ? = lim ww0 ?f(w) f(w 0) w w0 ? , ?w = z,w0= z0.?g(z)?z0?f(z)?z0?.? ?f(z)?f(z)?. 12.?,?w = f(z)?D?,?D?f(z) 6= 0. ?.?D?w?z?124

27、? ?6.1?, w?z? ?4?123?.?w = f(z)? z = f1(w).?z = f1(w)?G = f(z)|z D5?.?z0 D,?f1(w) ?w0= f(z0)?.?, lim ww0 f1(w) f1(w0) w w0 = lim zz0 z z0 f(z) f(z0) = lim zz0 1 f(z)f(z0) zz0 = 1 f(z0). ?z = f1(w)?G?,?C-R? x u = y v , x v = y u (w = u + iv,z = x + iy) ?.?. 13.?,?shz = ez ez 2 . 16.?7?. 17.?z? .? ? 2

28、.1?I, II, III?.?w(i) = i?,?w3= r 1 3ei +4 3 .?i = ei 2, = 2. ? w3(i) = ei 7 6= ei 6= 1 2( 3 + i). 18.(1) ez= 1 + i3 = 2ei 3, z = Ln(1 + i3) = ln2 + i(1 3 + 2k),k = 0,1,2,. (2) z = ei 2= i. 19.?z = rei,z 1 = e?, = q (rcos 1)2+ r2sin2 = 1 + r2 2rcos.? Reln(z 1) = ln = 1 2 ln(1 + r2 2rcos). 4 ?.? ?,?.?

29、 w?z?,?. 5 ?.? ?. 10? - O u 6 v # # cc c I II III ?2.1 20.? Ln(1 + i) = ln 2 + i(/4 + 2k), k = 0,1,2, . (1 + i)i= eiLn(1+i)= eiln 2 e(/4+2k)= e(/4+2k)cos(ln 2) + isin(ln2), k = 0,1,2, . 3i= eiLn3= eiln3+i2k= eiln3e2k= e2kcos(ln3) + isin(ln3), k = 0,1,2, . ii= eiLni= e(/2+2k), k = 0,1,2, . e2+i= e2(c

30、os1 + isin1). ? ?: e2+i= e(2+i)Lne= e(2+i)(1+2ki)= e22k+i(1+4k)= e22kei,k = 0,1,2, . ?,? ?. 21. lim zz0 f(z) (z) = lim zz0 f(z) f(z0) (z) (z0) = lim zz0 f(z)f(z0) zz0 (z)(z0) zz0 = f(z0) (z0). 22.?(sinz)= cosz,?lim z0 sinz z = d dz sinz|z=0= cos0 = 1. 23.?0 Argw 2.?6.3.1?. 24. ? ?ei2z?= ? ?e2xi(1+2y

31、)?= e2x. ? ? ?e z2? ? = ? ? ?e x2y2+2xyi? ? = ex 2y2. Re(e 1 z) = Re ? e x x2+y2+i y x2+y2 ? = e x x2+y2cos y x2+y2. ? 1. Z 1+i 0 (x y + ix2)dz = Z 1 0 it2(1 + i)dt = i 1 3 . 2.(1)?z = it,1 t 1. Z i i |z|dz = Z 1 1 |t|idt = 2i Z 1 0 tdt = i. (2)?z = eit,/2 t /2. Z i i |z|dz = Z /2 /2 ieitdt = 2i. (3

32、)?z = eit,/2 t 3/2. Z i i |z|dz = Z 3/2 /2 ieitdt = 2i. 3.?f(z) = z,?k= zk1,? Sn= n X k=1 zk1(zk zk1). ?k= zk,? e Sn= n X k=1 zk(zk zk1). ?, Sn? e Sn?,?(Sn+ e Sn)/2?.? ? 1 2(Sn +eSn) = 1 2 n X k=1 (z2 k z 2 k1) = 1 2( 2 2). ? Z C zdz = 1 2( 2 2). 4.(1)?i?i? ? ?x2+ iy2? 1, ?2,? ? ? ? ? Z i i (x2+ iy2

33、)dz ? ? ? ? 2. (2)?z = eit,/2 t /2,?x = cost,y = sint.? ? ? ?x2+ iy2?= p cos4t + sin4t = p 1 2cos2tsin2t 1. ?,? ? ? ?R i i(x 2 + iy2)dz ? ? ? . (3)?z = 2t + i,0 t 1.? ? ? ? ? 1 z2 ? ? ? ? = 1 |z|2 = 1 4t2+ 1 1. ? ? ? ? ? Z 2+i i dz z2 ? ? ? ? 2. 5.(1)?cosz?(n + 1/2),n = 0,1,2,?1/cosz? ?.? ? Z |z|=1

34、dz cosz = 0. (2)?z2+ 2z + 2?1 i,?1/(z2+ 2z + 2)? ?.? Z |z|=1 dz z2+ 2z + 2 = 0. 11 12? (3)? ez z2+ 5z + 6 ?2?3,? ?.? ? Z |z|=1 ez z2+ 5z + 6dz = 0. 6. Z |z|=1 dz z = Z 2 0 ieit eit dt = 2i. Z |z|=1 dz |z| = Z |z|=1 dz = 0. Z |z|=1 |dz| z = Z 2 0 1 eit dt = 0. Z |z|=1 |dz| |z| = Z 2 0 dt = 2. 7.? 0 =

35、 Z |z|=1 dz z + 2 = Z 2 0 sin + icos cos + 2 + isin d = Z 2 0 2sin + i(1 + 2cos) 5 + 4cos d. ? Z 2 0 1 + 2cos 5 + 4cos d = 0. ? Z 2 1 + 2cos 5 + 4cos d = Z 2 1 + 2cos(2 ) 5 + 4cos(2 )d t=2 = Z 0 1 + 2cost 5 + 4costdt. ? Z 0 1 + 2cos 5 + 4cos d = 0. 8.?f(z) = 2z2 z + 1,? ?. (1) 4i. (2) 6i. 9.(1) Z C

36、sin 4z z2 1dz = 2i sin 4z z 1 ? ? ? ? z=1 = 2 2 i. (2) Z C sin 4z z2 1dz = 2i sin 4z z + 1 ? ? ? ? z=1 = 2 2 i. (3)?,? 2i. 10.?|z| 3 ?, f(z) = 2i(3z2+ 7z + 1).?f(z) = 2i(6z + 7),|z| 3. ? f(1 + i) = 2(6 + 13i). 11.?, 2i = Z |z|=1 ez z dz = Z 2 0 ecossin(sin) + icos(sin)d. ? Z 2 0 ecoscos(sin)d = 2. ?

37、7?,? Z 2 ecoscos(sin)d = Z 0 ecoscos(sin)d. ? Z 2 0 ecoscos(sin)d = . ?13 12.?C?x2+ y2= 1?,? Z C F(z)dz = 0;?C?(x 2)2+ y2= 1?,? ? Z C F(z)dz = 2i z + 6 z + 2 ? ? ? ? z=2 = 4i;?C?(x + 2)2+ y2= 1?,? Z C F(z)dz = 2i z + 6 z 2 ? ? ? ? z=2 = 2i. 14. (1) Z C cosz (z 1)5 dz = 2i 4! d4cosz dz4 ? ? ? ? z=1 =

38、 2i 4! 4cos = 5i 12 . (2) i2sin(1 /4).?55?8. 15.? ?zn n! ?2 = 1 2i Z C znez n!n d , ? zn n! = 1 2i Z C ez n d . ?, 1 2i Z C ez n d = 1 2i 2i n! dnez dn ? ? ? ? =0 = 1 n! znez?=0= zn n! . 16.?,?: 1,?; 2,?; 3, ?.?.?z(t) = x(t) + iy(t),w(t) = fz(t) = ux(t),y(t) + ivx(t),y(t).? w(t) = d dtux(t),y(t) + i

39、 d dtvx(t),y(t) = uxx (t) + uyy(t) + ivxx(t) + ivyy(t). ?f(z) = ux+ ivx?C-R?,? w(t) = uxx(t) vxy(t) + ivxx(t) + uxy(t) = (ux+ ivx)x(t) + iy(t) = f(z)z(t). ?f(z) 6= 0,z(t) 6= 0,?w(t) 6= 0.?f(z)?,?f(z)?z?.? ?,?fz(t)?t?.?w(t)?t?,?.?,? ?f(z)? ?,?f(z)?,?C?,?.? ?. 17. Z (w)dw = Z w(t)w(t)dt = Z f(z(t)fz(t

40、)z(t)dt = Z C f(z)f(z)dz. ? 1.(1)lim n n r 1 nn = lim n 1 n = 0.?. (2)lim n n nn = lim n n = .?0. (4)lim n ?n + 1 n ?k = 1.?1. X n=0 nkzn?.? ? lim n |nkzn| = lim n nk= . (3)?(5)?75?4. 2.(1) 1 az + b = 1 b 1 1 + a b z = 1 b X n=0 ? a b ?n zn,? ? ?b a ? ? ?. (2)?ez 2 = X n=0 z2n n! ,? Z z 0 ez 2dz = X

41、 n=0 1 n! Z z 0 z2ndz = X n=0 z2n+1 (2n + 1)n!. ? Z z 0 ez 2dz ? ?,?. (3)? sinz z = X n=0 (1)nz2n (2n + 1)! ,? Z z 0 sinz z dz = X n=0 (1)n (2n + 1)! Z z 0 z2ndz = X n=0 (1)nz2n+1 (2n + 1)(2n + 1)!. ?,?,?.?lim n n n! = ,?lim n 2n+1 q 1 (2n+1)(2n+1)! = 0.?.? Z z 0 sinz z dz? ?.? (4)?: cos2z = 1 2(1 +

42、 cos2z) = 1 2 + 1 2 X n=0 (1)n(2z)2n (2n)! = 1 + 1 2 X n=1 (1)n(2z)2n (2n)! . ?:?(cos2z)= 2sinz cosz = sin2z = X n=0 (1)n(2z)2n+1 (2n + 1)! ?, cos2z =1 Z z 0 sin2zdz = 1 X n=0 (1)n (2n + 1)! Z z 0 (2z)2n+1dz =1 1 2 X n=0 (1)n(2z)2n+2 (2n + 2)! = 1 + 1 2 X n=1 (1)n(2z)2n (2n)! . ?cos2z?,?. (5)?: sin2

43、z = 1 2(1 cos2z) = 1 2 1 2 X n=0 (1)n(2z)2n (2n)! = 1 2 X n=1 (1)n(2z)2n (2n)! . ?:?(sin2z)= 2sinz cosz = sin2z = P n=0 (1)n(2z)2n+1 (2n+1)! ?, sin2z = Z z 0 sin2zdz = X n=0 (1)n (2n + 1)! Z z 0 (2z)2n+1dz =1 2 X n=0 (1)n(2z)2n+2 (2n + 2)! = 1 2 X n=1 (1)n(2z)2n (2n)! . 15 16? ?sin2z?,?. ?:?, (4)?(5

44、)?.? sin2z + cos2z = 1, cos2z sin2z = cos2z = X n=0 (1)n(2z)2n (2n)! ? cos2z = 1 2 + 1 2 X n=0 (1)n(2z)2n (2n)! = 1 + 1 2 X n=1 (1)n(2z)2n (2n)! , sin2z = 1 2 1 2 X n=0 (1)n(2z)2n (2n)! = 1 2 X n=1 (1)n(2z)2n (2n)! . (6) 1 (1 z)2 = ? 1 1 z ? = d dz X n=0 zn= X n=1 nzn1= X n=0 (n + 1)zn,|z| 1. 3.(1)l

45、im n n p qn 2 = lim n qn= 0.?. (2)?,?ck? ck= ( 1,?k = n!,n = 0,1,2, , 0,?. ? lim k k c k= 1. ?1. (3)?,?cn? cn= ( 4n,n = 0,2,4, , 2n,n = 1,3,5, . ? lim n n c n= 4. ?1/4. (4) cn= n!/nn.? lim n cn+1 cn = lim n (n + 1)! (n + 1)n+1 nn n! = lim n ? 1 1 + 1/n ?n = e1. ?e. 4. ezln(1 + z) = ? 1 + z + z2 2 +

46、z3 6 + z4 24 + ? z z2 2 + z3 3 z4 4 + z5 5 ? = z + ? 1 1 2 ? z2+ ?1 2 1 2 + 1 3 ? z3+ ?1 6 1 4 + 1 3 1 4 ? z4 + ? 1 24 1 12 + 1 6 1 4 + 1 5 ? z5+ = z + 1 2z 2 + 1 3z 3 + 3 40z 5 + ,|z| 1. ?17 5.(1)?:?cos(n)z = cos(z + n 2 ),?cos(n)1 = cos(1 + n 2 ),? cosz = X n=0 cos(1 + n 2 ) n! (z 1)n,|z 1| . ?: c

47、osz = cos1 + (z 1) = cos1cos(z 1) sin1sin(z 1) = cos1 X n=0 (1)n(z 1)2n (2n)! sin1 X n=0 (1)n(z 1)2n+1 (2n + 1)! = X n=0 cos1cos n 2 n! (z 1)n X n=0 sin1sin n 2 n! (z 1)n = X n=0 cos1cos n 2 sin1sin n 2 n! (z 1)n = X n=0 cos ? 1 + n 2 ? n! (z 1)n. (2)?sin(n)z = sin(z + n 2 ),?sin(n)1 = sin(1 + n 2 )

48、,? sinz = X n=0 sin(1 + n 2 ) n! (z 1)n,|z 1| . (3) z z + 2 = 1 2 z + 2 = 1 2 3 + (z 1) = 1 2 3 1 1 + z 1 3 = 1 2 3 X n=0 (1)n (z 1)n 3n , |z 1| 3. (4) z z2 2z + 5 = z (z 1 + 2i)(z 1 2i) = 1 4i ? z z 1 2i z z 1 + 2i ? = 1 4i ? 1 + 2i z 1 2i 1 2i z 1 + 2i ? = 1 8 1 + 2i 1 z 1 2i + 1 2i 1 + z 1 2i = 1

49、 8 (1 + 2i) X n=0 1 (2i)n (z 1)n+ (1 2i) X n=0 (1)n (2i)n (z 1)n # = 1 4 X n=0 1 (2i)2n (z 1)2n+ 2i X n=0 1 (2i)2n+1 (z 1)2n+1 # = 1 4 X n=0 1 (4)n (z 1)2n+ X n=0 1 (4)n (z 1)2n+1 # = 1 4 X n=0 cos n 2 4n (z 1)n+ X n=0 sin n 2 4n (z 1)n = 2 4 X n=0 sin ?n 2 + 4 ? 4n (z 1)n,|z 1| 2. 18? 6.?f(z) = 1 1 z z2 ? 1 5 2 ,?|z| n?, z0?mn?;?m = n?, z0?;?m n?, z0?nm?. 9.(1)?0 |z| 1?, z + 1 z2(z 1) = ? 1 z2 + 1 z ? X n=0 zn= 1 z2 2 X n=0 zn1; ?1 |z| ?, z + 1 z2(z 1) = ? 1 z3 + 1 z2 ? X n=0 1 zn = 1 z2 + 2