Question

e^2iz展开成z的幂级数

Answer 1

问：e^2iz展开成z的幂级数

答：ez2展开成z的幂级数：可以用乘积展开，也可以先变形再展开。e^(z/(z-1))=e^(1+1/(z-1))可以按照泰勒展开令[e^(1+1/(z-1))](n)'代表n次导数那么[e^(1+1/(z-1))](1)'=[e^(1+1/(z-1))]*[-1/(z-1)^2][e^(1+1/(z-1))](2)'=[e^(1+1/(z-1))]*[1/(z-1)^4]+[e^(1+1/(z-1))]*[2/(z-1)^3]=[e^(1+1/(z-1))]*[(2z-1)/(z-1)^4][e^(1+1/(z-1))](3)'= (e^(z/(z-1)) (-6 z^2+6 z-1))/(z-1)^6