展开 (e^{ipiθ}-e^{-ipiθ})^6

解答

展开

(e^{iπ θ} - e^{- iπ θ})^{6}

- 64 sin^{6} (π θ)

求解步骤

(e^{iπ θ} - e^{- iπ θ})^{6}

e^{iπ θ} = cos (π θ) + i sin (π θ)

e^{- iπ θ} = cos (π θ) - i sin (π θ)

= (cos (π θ) + i sin (π θ) - (cos (π θ) - i sin (π θ)))^{6}

乘开

cos (π θ) + sin (π θ) i - (cos (π θ) - i sin (π θ)) : 2 i sin (π θ)

= (2 i sin (π θ))^{6}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{6} i^{6} sin^{6} (π θ)

2^{6} = 64

= 64 i^{6} sin^{6} (π θ)

i^{6} = - 1

= 64 (- 1) sin^{6} (π θ)

整理后得

= - 64 sin^{6} (π θ)

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