数学物理方法答案83144

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,℄(-m48TAAkG
q,BYt4Zg℄6Zg,m'do$Z21o$Z24Fo$Z211oZ214VoZ1-6222

2、222>Tl(x+y)=(x−y)+4xy=a+b,4"Ax+y=a+b.-(1.1)qs)#
l√x2=1(a+a2+b2),2√y2=1(−a+a2+b2).2>~
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(1.1)qs#.slfb6=0,s√s√√a+a2+b2b−a+a2+b2a+bi=±+i.2

3、b

4、2√√Eb=0ml,fa≥0,ei^±a;fa<0,ei^±i−a.√1−2i−2−i=16+8i,1688512.(1)3−4i5i253^25,3^25,^25,w^arctan2+2kπ,k=0,±1,

5、±2,···.√nπnnπ1+3iiinπnπ(2)33=cos>2=e=e3+isin3,n=2,3,4e,tlo}'1.)2,√13√!n−+i,n=2;1+3i22=−1,n=3;2√13−−i,n=4.22√1,32πA-n=2,3^−23^2,w^3+2kπ,k=0,±1,±2,···;n=3√,3^−1,133^0,w^(2k+1)π,k=0,±1,±2,···;n=4,3^−2,3^−2,w^4π+2kπ,k=0,±1,±2,···.3(3)*

6、.22√π√√π/4+2nπq√i4,4i2,n=0,1.1+2:U,4q^1+i=2eA-1+i=2eA-n=0,3q^2,√√√−1+24π1+23^q2,^2,w^8+2kπ,k=0,±1,±2,···;n=1,3^−2,3√√−1+249π^−,^2,w^+2kπ,k=0,±1,±2,···.28√π√π√3+i=2ei6,−3−3−i2=−i−31(4)4;(3+i)=2e8.A-(3+i)m3^0Æ3^−8,1π^8,w^−2+2kπ,k=0,±1,±2,···.√(5)13π3^2,3^−2,^1,w^−3+2kπ,k=0,±1,±

7、2,···.iπ−iπiπππz11i5π15π5π=e4,z6,12=2(cos12=3.4z12=2e;z1z2=2e12+isin12),z=2e2(cos12+isin12).22211212124.>z+=2cosθl,z+=z+2+=4cosθ.o#1$qw4lz−=−4sinθ.zzz2z1=±2isinθ.±iθA-z−z>Tlz=e.A-m1±imθ∓imθz+=e+e=2cosmθ.zm12−2−1e−iπ2πi2π5.(1)√√3>z=1+3i=13=iπ=eeargz=3.+ie322√π√3−i=2e−i6,66−iπ6i

8、π(2)4;z=(3−i)=2e=2e,A-argz=π.√√ππππ√i−iii6.(1)i(1−3i)(3+i)=e2·2e3·2e6=4e3=2+23i.(2)iπ/2√√5i5ei(π/2−arctan1)π1π1=√=5e2=5cos(−arctan)+isin(−arctan)iarctan12+i5e22222√11√12=5sin(arctan)+icos(arctan)=5√+i√=1+2i.2255√7i7·3π√7i5π√7√√(3)(−1+i)7=2e4=2e4=2(−2−i2)=−8(1+i).22√10π2π√(4)(1+3i)−10

9、=2−10e−i3=2−10ei3=2−11(−1+3i).7.444iπ>z=−a=ael4*/^(2n+1)πizn=ae4,n=0,1,2,3.√√√√z=2a(1+i),z=2a(−1+i),z=2a(−1−i),z=2a(1−i).021222328.(1)z1=x1+iy1,z2=x2+iy2,VRe(z1+z2)=x1+x2=Re(z1)+Re(z2).tz1=i,z2=1+i,VRe(z1z2)=−16=0=Re(z1)Re(z2).A-Re(z1z2)6=Re(z1)Re(z2)).E,.(2