let-f-z-z-z-2-z-2-developp-f-at-integr-serie-

Question Number 39038 by maxmathsup by imad last updated on 01/Jul/18

l e t f (z) = \frac{z}{z^{2} - z + 2}

d e v e l o p p f a t i n t e g r s e r i e .

Commented by prof Abdo imad last updated on 02/Jul/18

l e t d e c o m p o s e f (x) o n d i d e C (x)

Δ = 1 - 4 (2) = - 7 = {(i \sqrt{7})}^{2} \Rightarrow

z_{1} = \frac{1 + i \sqrt{7}}{2} a n d z_{2} = \frac{1 - i \sqrt{7}}{2} \Rightarrow

f (z) = \frac{z}{(z - z_{1}) (z - z_{2})} = \frac{a}{z - z_{1}} + \frac{b}{z - z_{2}}

a = \frac{z_{1}}{z_{1} - z_{2}} = \frac{z_{1}}{i \sqrt{7}} = \frac{- {i z}_{1}}{\sqrt{7}}

b = \frac{z_{2}}{z_{2} - z_{1}} = \frac{z_{2}}{- i \sqrt{7}} = \frac{{i z}_{2}}{\sqrt{7}} \Rightarrow

f (x) = \frac{- {i z}_{1}}{\sqrt{7} (z - z_{1})} + \frac{{i z}_{2}}{\sqrt{7} (z - z_{2})} \Rightarrow

f^{(n)} (x) = \frac{- {i z}_{1}}{\sqrt{7}} {\frac{{(- 1)}^{n} n!}{{(z - z_{1})}^{n + 1}}} + \frac{{i z}_{2}}{\sqrt{7}} {\frac{{(- 1)}^{n} n!}{{(z - z_{2})}^{n + 1}}}

= \frac{{(- 1)}^{n} n!}{\sqrt{7}} {\frac{z_{2}}{{(z - z_{2})}^{n + 1}} - \frac{z_{1}}{{(z - z_{1})}^{n + 1}}} \Rightarrow

f^{(n)} (0) = \frac{{(- 1)}^{n} n!}{\sqrt{7}} {\frac{z_{2}}{{(- 1)}^{n + 1} z_{2}^{n + 1}} - \frac{z_{1}}{{(- 1)}^{n + 1} z_{1}^{n + 1}}}

= \frac{n!}{\sqrt{7}} {\frac{- z_{2}}{z_{2}^{n + 1}} + \frac{z_{1}}{z_{1}^{n + 1}}} f i n a l l y

f (z) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (0)}{n!} z^{n} = \sum_{n = 0}^{\infty} \frac{1}{\sqrt{7}} (\frac{1}{z_{1}^{n}} - \frac{1}{z_{2}^{n}}) z^{n}

Commented by prof Abdo imad last updated on 02/Jul/18

l e t s i m p l i f y \frac{1}{z_{1}^{n}} - \frac{1}{z_{2}^{n}}

= \frac{z_{2}^{n} - z_{1}^{n}}{{(z_{1} z_{2})}^{n}} = \frac{{(\frac{1 - i \sqrt{7}}{2})}^{n} - {(\frac{1 + i \sqrt{7}}{2})}^{n}}{2}

= - \frac{1}{2^{n + 1}} {{(1 + i \sqrt{7})}^{n} - {(1 - i \sqrt{7})}^{n}}

= - \frac{1}{2^{n + 1}} {\sum_{k = 0}^{n} C_{n}^{k} {(i \sqrt{7})}^{k} - \sum_{k = 0}^{n} C_{n}^{k} {(- i \sqrt{7})}^{k}}

= \frac{- 1}{2^{n + 1}} \sum_{k = 0}^{n} C_{n}^{k} {{(i \sqrt{7})}^{k} - {(- i \sqrt{7})}^{k}}

= \frac{- 1}{2^{n + 1}} \sum_{p = 0}^{[\frac{n - 1}{2}]} C_{n}^{2 p + 1} {2 i {(- 1)}^{p} 7^{p} \sqrt{7}}

= \frac{- i}{2^{n}} \sqrt{7} \sum_{p = 0}^{[\frac{n - 1}{2}]} C_{n}^{2 p + 1} {(- 7)}^{p} \Rightarrow

f (z) = \sum_{n = 0}^{\infty} \frac{- i}{2^{n}} {\sum_{p = 0}^{[\frac{n - 1}{2}]} {(- 7)}^{p} C_{n}^{2 p + 1}} z^{n}