展开 (1+(x+iy)^3)^2

解答

展开

(1 + (x + i y)^{3})^{2}

(1 + 2 x^{3} - 6 x y^{2} + x^{6} - 15 x^{4} y^{2} + 15 x^{2} y^{4} - y^{6}) + (6 x^{2} y - 2 y^{3} + 6 x^{5} y - 20 x^{3} y^{3} + 6 x y^{5}) i

求解步骤

(1 + (x + i y)^{3})^{2}

使用完全平方公式:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

(1 + (x + i y)^{3})^{2} = 1^{2} + 2 \cdot 1 \cdot (x + i y)^{3} + ((x + i y)^{3})^{2}

= 1^{2} + 2 \cdot 1 \cdot (x + i y)^{3} + ((x + i y)^{3})^{2}

化简

1^{2} + 2 \cdot 1 \cdot (x + i y)^{3} + ((x + i y)^{3})^{2} : 1 + 2 (x^{3} + 3 x^{2} y i - 3 x y^{2} - i y^{3}) + x^{6} + 6 x^{5} y i - 15 x^{4} y^{2} - 20 x^{3} i y^{3} + 15 x^{2} y^{4} + 6 x i y^{5} - y^{6}

= 1 + 2 (x^{3} + 3 x^{2} y i - 3 x y^{2} - i y^{3}) + x^{6} + 6 x^{5} y i - 15 x^{4} y^{2} - 20 x^{3} i y^{3} + 15 x^{2} y^{4} + 6 x i y^{5} - y^{6}

展开

2 (x^{3} + 3 x^{2} y i - 3 x y^{2} - i y^{3}) : 2 x^{3} + 6 x^{2} y i - 6 x y^{2} - 2 i y^{3}

= 1 + 2 x^{3} + 6 x^{2} y i - 6 x y^{2} - 2 i y^{3} + x^{6} + 6 x^{5} y i - 15 x^{4} y^{2} - 20 x^{3} i y^{3} + 15 x^{2} y^{4} + 6 x i y^{5} - y^{6}

Rewrite in standard complex form:

(1 + 2 x^{3} - 6 x y^{2} + x^{6} - 15 x^{4} y^{2} + 15 x^{2} y^{4} - y^{6}) + (6 x^{2} y - 2 y^{3} + 6 x^{5} y - 20 x^{3} y^{3} + 6 x y^{5}) i

= (1 + 2 x^{3} - 6 x y^{2} + x^{6} - 15 x^{4} y^{2} + 15 x^{2} y^{4} - y^{6}) + (6 x^{2} y - 2 y^{3} + 6 x^{5} y - 20 x^{3} y^{3} + 6 x y^{5}) i

e x p an d 2 s^{2} e^{2 t} (1 - 3 s^{3} e^{- 3 t}) + 9 s^{3} e^{t}

e x p an d (3 - \frac{x}{47829669})^{20}

e x p an d e^{x t - t} \cdot (U e^{x t} - Ux t - U)

e x p an d (6 e^{x} - e^{2 x} - 9) (- 3 2 e^{x} - 6 x + 5)

e x p an d t (e^{- 3 t} + e^{- 2 t})