[複變] 複數方程
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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