急求几道复变函数题。
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题目
1)i的i此方的值
2)cosi的值
3)Ln(-i)的值
4)(√3+i)^4
5)(1+i)^(1/4)
6)e^(3+i4)
7)ln(3+i4)=a+ib
8)sin(5i)
答案
1)i^i=e^(iLni)=e^(ii[2kPi+Pi/2])=e^(-2k-1/2)Pi
2)cosi=[e^(ii)+e^(-ii)]/2=[e^(-1)+e]/2
3)Ln(-i)=ln|-i|+i[2kPi+arg(-i)]=i[2k-1/2]Pi
4)根据z^n=ρ^n(cos(nθ)+sin(nθ))
ρ^n=√((√3)^2+1^2)^4=16
nθ=4arctan1/√3=2π/3
(√3+i)^4=16(cos(2π/3)+sin(2π/3))=-8+i8√3
5) 根据z^(1/n)=ρ^(1/n)(cos(θ/n)+sin(θ/4))
ρ^(1/n)=(√2)^(1/4)=2^(1/16)
θ/n =[arctan(1/1)]/4=π/16
(1+i)^(1/4)=2^(1/16)cos(π/16)+i2^(1/16)sin(π/16)
6)e^(3+i4)=e^3*e^(i4)=e^3cos(arctan4)+ie^3sin(arctan4)
7)ln(3+i4)=a+ib
3+i4 =e^(a+ib)=e^a*e^(ib)
e^a=sqrt(3^2+4^2)=5
a=ln5
b=Arctan(4/3)
=>ln(3+i4)=ln5+iArctan(4/3)
8)根据sinz=1/(i2)(e^(iz)-e^(-iz))展开
sin(5i) =1/(i2)(e^(-5)-e^(5))
=-i/2(e^(-5)-e^(5))
不知道你是要求解你的题还是需要几道题,希望能帮助到你
1)i的i此方的值
2)cosi的值
3)Ln(-i)的值
4)(√3+i)^4
5)(1+i)^(1/4)
6)e^(3+i4)
7)ln(3+i4)=a+ib
8)sin(5i)
答案
1)i^i=e^(iLni)=e^(ii[2kPi+Pi/2])=e^(-2k-1/2)Pi
2)cosi=[e^(ii)+e^(-ii)]/2=[e^(-1)+e]/2
3)Ln(-i)=ln|-i|+i[2kPi+arg(-i)]=i[2k-1/2]Pi
4)根据z^n=ρ^n(cos(nθ)+sin(nθ))
ρ^n=√((√3)^2+1^2)^4=16
nθ=4arctan1/√3=2π/3
(√3+i)^4=16(cos(2π/3)+sin(2π/3))=-8+i8√3
5) 根据z^(1/n)=ρ^(1/n)(cos(θ/n)+sin(θ/4))
ρ^(1/n)=(√2)^(1/4)=2^(1/16)
θ/n =[arctan(1/1)]/4=π/16
(1+i)^(1/4)=2^(1/16)cos(π/16)+i2^(1/16)sin(π/16)
6)e^(3+i4)=e^3*e^(i4)=e^3cos(arctan4)+ie^3sin(arctan4)
7)ln(3+i4)=a+ib
3+i4 =e^(a+ib)=e^a*e^(ib)
e^a=sqrt(3^2+4^2)=5
a=ln5
b=Arctan(4/3)
=>ln(3+i4)=ln5+iArctan(4/3)
8)根据sinz=1/(i2)(e^(iz)-e^(-iz))展开
sin(5i) =1/(i2)(e^(-5)-e^(5))
=-i/2(e^(-5)-e^(5))
不知道你是要求解你的题还是需要几道题,希望能帮助到你
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