数学物理方法，Z的四次方加一等于零，用指数式计算

Question

数学物理方法，Z的四次方加一等于零，用指数式计算拜托了

Answer 1

在实数域无解！只能在复数域求解：

Z⁴ + 1 = 0 ----------------------- (1)

设：Z = R(cos a + j sin a) ------- (2)

Z⁴ = -1 = (cos π + j sin π) ------ (3)

Z  = cos[(π+2kπ)/n + j sin[(π+2kπ)/n] ---------- (4)     //: n=4，k=0，1，2，3 

Z₀ = cos(π/4)+jsin(π/4) =√2(1+j)/2 --- (5)               //: 验证：Z₀^4=-1

Z₁ = cos (3π/4)+jsin(3π/4) ------------ (6)

Z₂ = cos (5π/4)+jsin(5π/4) ------------ (7)

Z₃ = cos(7π/4)+jsin(7π/4) -------------(8)

(5,6,7,8)是4个复根。