find-the-taylor-series-of-f-z-sinz-z-pi-4-in-complex-number-

Question Number 147569 by Sozan last updated on 21/Jul/21

f i n d t h e t a y l o r s e r i e s o f f (z) = s i n z, z = \frac{π}{4} i n c o m p l e x n u m b e r

Answered by mathmax by abdo last updated on 22/Jul/21

f (z) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (\frac{π}{4})}{n!} {(z - \frac{π}{4})}^{n} we have f^{(n)} (z) = \sin (z + \frac{n π}{2}) \Rightarrow

f^{(n)} (\frac{π}{4}) = \sin (\frac{π}{4} + \frac{n π}{2}) = \sin (\frac{(2 n + 1) π}{4}) \Rightarrow

sinz = \sum_{n = 0}^{\infty} \frac{1}{n!} \sin (\frac{(2 n + 1) π}{4}) {(z - \frac{π}{4})}^{n}