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about De Moivre’s Theorem
發問:
Is the De Moivre’s Theorem (cos θ + i sin θ)^n = cos nθ + i sin nθ suitable for n is an irrational number? Explain in details. A math書，Pure Math書同埋老師都無講。
De Moivre's Theorem As a more complicated example of the value of the polar form, we'll prove De Moivre's theorem: 圖片參考：http://ccrma-www.stanford.edu/~jos/mdft/img217.png Working this out using sum-of-angle identities from trigonometry is laborious (see §3.13 for details). However, using Euler's identity, De Moivre's theorem simply ``falls out'': 圖片參考：http://ccrma-www.stanford.edu/~jos/mdft/img218.png Moreover, by the power of the method used to show the result, 圖片參考：http://ccrma-www.stanford.edu/~jos/mdft/img69.png can be any real number, not just an integer. so if you using Euler's identity, the result is suitable for n is an irrational number is not difficult to see
其他解答:
about De Moivre’s Theorem
發問:
Is the De Moivre’s Theorem (cos θ + i sin θ)^n = cos nθ + i sin nθ suitable for n is an irrational number? Explain in details. A math書，Pure Math書同埋老師都無講。
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最佳解答:De Moivre's Theorem As a more complicated example of the value of the polar form, we'll prove De Moivre's theorem: 圖片參考：http://ccrma-www.stanford.edu/~jos/mdft/img217.png Working this out using sum-of-angle identities from trigonometry is laborious (see §3.13 for details). However, using Euler's identity, De Moivre's theorem simply ``falls out'': 圖片參考：http://ccrma-www.stanford.edu/~jos/mdft/img218.png Moreover, by the power of the method used to show the result, 圖片參考：http://ccrma-www.stanford.edu/~jos/mdft/img69.png can be any real number, not just an integer. so if you using Euler's identity, the result is suitable for n is an irrational number is not difficult to see
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