求解方程z4+16=0(复数运算）要过程

Question

求解方程z4+16=0(复数运算）要过程

Answer 1

z^4+16=0
那么:
z^4=-16=16e^[i(π+2kπ)]
那么z=16^(1/4) * e^[i(π+2kπ)/4] (k=0,1,2,3)
=2 * e^[i(π+2kπ)/4]
则：
再运用欧拉公式 e^(iθ)=cosθ+isinθ
z1=2e^(iπ/4)=2(cosπ/4+isinπ/4)=√2(1+i)
z2=2e^(i3π/4)=2(cos3π/4+isin3π/4)=√2(-1+i)
z3=2e^(i5π/4)=2(cosπ5/4+isin5π/4)=√2(-1-i)
z4=2e^(i7π/4)=2(cosπ7/4+isin7π/4)=√2(1-i)