Exercise 7.27: Finding more series

Find power series for each of the following functions:

b) $\displaystyle \cos(z^2)$

c) $\displaystyle z^2\sin(z)$

d) $\displaystyle (\sin(z))^2$

You might focus on the first few terms for that last one.

For b.) $$\sin(z)=\sum_{n=0}^\infty(-1)^{n+1}\frac{z^{2n+1}}{(2n+1)!}$$
$$f(z)=z^2\sin(z)=z^2\sum_{n=0}^\infty(-1)^{n+1}\frac{z^{2n+1}}{(2n+1)!}=\sum_{n=0}^\infty(-1)^{n+1}\frac{z^{2n+1}}{(2n+1)!}=\sum_{n=0}^\infty(-1)^{n+1}\frac{z^{2n+3}}{(2n+1)!}$$

B) Find the power series representation about the origin for $\cos(z^2)$.

Note that the power series about the origin for $\cos(z)=\displaystyle\sum_{k=0}^{\infty}(-1)^{2k}\frac{z^{2k}}{(2k)!}$.

If we instead evaluate $\cos(z^2)$, we have $\cos(z^2)=\displaystyle\sum_{k=0}^{\infty}(-1)^{2k}\frac{(z^2)^{2k}}{(2k)!}=\displaystyle\sum_{k=0}^{\infty}(-1)^{2k}\frac{z^{4k}}{(2k)!}$.