[複變] 複數方程

[sulanpa (...)]

Solve the equation (z+1)^4 = z^4 where z is a complex number.

我的做法:

let z = x + iy  x,y屬於R

=> (x+1+iy)^4 = (x+iy)^4

=> [(x+1+iy)/(x+iy)]^4 = 1

=> [(x^2+x+y^2-iy)/(x^2+y^2)]^4 = 1

let w = [(x^2+x+y^2)/(x^2+y^2)] + i[-y/(x^2+y^2)]

=> w^4 = 1 = cos0 + isin0

=> w = cos(kpi/2) + isin(kpi/2) for k = 0,±1,±2,...

if k = 0, then w = 1

=> (x^2+x+y^2)/(x^2+y^2) = 1 and -y/(x^2+y^2) = 0

=> x,y無實數解

if k = 1, then w = i

=> (x^2+x+y^2)/(x^2+y^2) = 0 and -y/(x^2+y^2) = 1

=> x = -1/2 , y = -1/2

if k = 2, then w = -1

=> (x^2+x+y^2)/(x^2+y^2) = -1 and -y/(x^2+y^2) = 0

=> x = -1/2 , y = 0

if k = 3, then w = -i

=> (x^2+x+y^2)/(x^2+y^2) = 0 and -y/(x^2+y^2) = -1

=> x = -1/2 , y = 1/2

共解出三個相異的z

但複數方程不是冪次多少就解出多少個根嗎?

請問有哪個步驟的邏輯錯了嗎?

還是哪裡解錯了呢?


謝謝

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