Answer to Question #315435 in Complex Analysis for Roshan

Question

Obtain first three terms of taylors series for f(z) = sin z about z =0

Answer 1

$f\left( 0 \right) =\sin 0=0\\f'\left( 0 \right) =\cos z|_{z=0}=1\\f''\left( 0 \right) =-\sin z|_{z=0}=0\\f'''\left( 0 \right) =-\cos z|_{z=0}=-1\\f'^v\left( 0 \right) =\sin z|_{z=0}=0\\f^v\left( 0 \right) =\cos z|_{z=0}=1\\First\,\,3terms:\\f\left( z \right) =\sum_{n=0}^5{\frac{1}{n!}f^{\left( n \right)}\left( z \right) z^n}+o\left( z^5 \right) =z-\frac{1}{3!}z^3+\frac{1}{5!}z^5+o\left( z^5 \right) =\\=z-\frac{1}{6}z^3+\frac{1}{120}z^5+o\left( z^5 \right)$