求f(z)=(1-e^2z)/z^2在0

Question

求f(z)=(1-e^2z)/z^2 在0

Answer 1

e^z = ∑{0 ≤ n} z^n/n!,
故e^(2z) = ∑{0 ≤ n} (2z)^n/n!= ∑{0 ≤ n} 2^n/n!·z^n,
进而得1-e^(2z) = -∑{1 ≤ n} 2^n/n!·z^n.
于是(1-e^(2z))/z^2 = -∑{1 ≤ n} 2^n/n!·z^(n-2) = = -∑{-1 ≤ n} 2^(n+2)/(n+2)!·z^n.
这就是(1-e^(2z))/z^2以原点为中心的Laurent展开.