将e∧zcosz在o点展开到C5z∧5（复变函数的泰勒级数）急求啊!

Question

将e∧zcosz在o点展开到C5z∧5（复变函数的泰勒级数）急求啊!

Answer 1

e^zcosz=e^z[e^iz+e^(-iz)]/2
=[e^(1+i)z+e^(1-i)z]/2
由e^x=1+x+x^2/2+x^3/6+x^4/4!+.
e^(1+i)z=1+(1+i)z+(1+i)^2z^2/2+(1+i)^3z^3/6+(1+i)^4z^4/24+(1+i)^5z^5/120+.
e^(1-i)z=1+(1-i)z+(1-i)^2z^2/2+(1-i)^3z^3/6+(1-i)^4z^4/24+(1-i)^5z^5/120+.
两式相加,再除以2得：
e^zcosz=1+z-z^3/3-z^4/6-z^5/30+.