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单词 LimitForExpz
释义

limit for exp(z)


For any complex numberMathworldPlanetmathPlanetmath z, we have

limn(1+zn+o(1n))n=expz,

where exp denotes the exponential functionDlmfDlmfMathworldPlanetmathPlanetmath.
Proof:For α0, we have

ln(1+α)=k=1(-1)k-1αkk
=α+O(α2).

Therefore

(1+zn+o(1n))n=exp(nln(1+zn+o(1n)))
=exp(n(zn+o(1n)+O(1n2)))
=exp(z+o(1)+O(1n))expzforn.
随便看

 

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更新时间:2025/5/4 15:57:08