Question Number 57922 by maxmathsup by imad last updated on 14/Apr/19
$${decompose}\:{inside}\:{C}\left({x}\right)\:{the}\:{fraction}\:{F}\left({x}\right)\:=\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{{n}} }\:\:{with}\:{n}\:{integr}\:{natural} \\ $$$${and}\:{n}\geqslant\mathrm{1} \\ $$
Commented by maxmathsup by imad last updated on 30/Apr/19
$${we}\:{have}\:{F}\left({x}\right)\:=\frac{\mathrm{1}}{\left({x}−{i}\right)^{{n}} \left({x}+{i}\right)^{{n}} }\:=\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{{a}_{{k}} }{\left({x}−{i}\right)^{{k}} }\:+\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{{b}_{{k}} }{\left({x}+{i}\right)^{{k}} } \\ $$$${from}\:{another}\:{side}\:\:{ch}.\:{x}−{i}\:={t}\:{give}\:{F}\left({x}\right)\:={G}\left({t}\right)\:=\frac{\mathrm{1}}{{t}^{{n}} \left({t}+\mathrm{2}{i}\right)^{{n}} } \\ $$$${let}\:{determine}\:{D}_{{n}−\mathrm{1}} \left(\mathrm{0}\right)\:{for}\:\:{u}\left({t}\right)\:=\frac{\mathrm{1}}{\left({t}+\mathrm{2}{i}\right)^{{n}} }\:\Rightarrow{u}\left({t}\right)\:=\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\frac{{u}^{\left({k}\right)} \left(\mathrm{0}\right)}{{k}!}\:{t}^{{k}} \:+\frac{{t}^{{n}} }{{n}!}\xi\left({t}^{{n}} \right) \\ $$$${we}\:{have}\:{u}\left({t}\right)\:=\left({t}+\mathrm{2}{i}\right)^{−{n}} \:\Rightarrow{u}^{\left(\mathrm{1}\right)} \left({t}\right)\:=\left(−{n}\right)\left({t}+\mathrm{2}{i}\right)^{−{n}−\mathrm{1}} \:\Rightarrow \\ $$$${u}^{\left(\mathrm{2}\right)} \left({t}\right)=\left(−\mathrm{1}\right)^{\mathrm{2}} {n}\left({n}+\mathrm{1}\right)\:\left({t}+\mathrm{2}{i}\right)^{−{n}−\mathrm{2}} \:\Rightarrow{u}^{\left({p}\right)} \left({t}\right)\:=\left(−\mathrm{1}\right)^{{p}} {n}\left({n}+\mathrm{1}\right)…\left({n}+{p}−\mathrm{1}\right)\left({t}+\mathrm{2}{i}\right)^{−{n}−{p}} \\ $$$$\Rightarrow{u}^{\left({k}\right)} \left(\mathrm{0}\right)\:=\left(−\mathrm{1}\right)^{{k}} {n}\left({n}+\mathrm{1}\right)…\left({n}+{k}−\mathrm{1}\right)\left(\mathrm{2}{i}\right)^{−{n}−{k}} \:\Rightarrow \\ $$$${u}\left({t}\right)\:=\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} {n}\left({n}+\mathrm{1}\right)….\left({n}+{k}−\mathrm{1}\right)}{{k}!\left(\mathrm{2}{i}\right)^{{n}+{k}} }\:{t}^{{k}} \:+\frac{{t}^{{n}} }{{n}!}\:\xi\left({t}^{{n}} \right)\:\Rightarrow \\ $$$$\frac{{u}\left({t}\right)}{{t}^{{n}} }\:=\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} {n}\left({n}+\mathrm{1}\right)….\left({n}+{k}−\mathrm{1}\right)}{\left(\mathrm{2}{i}\right)^{{n}+{k}} {k}!\:{t}^{{n}−{k}} }\:+\frac{\mathrm{1}}{{n}!}\:\xi\left(\mathrm{1}\right)\:\left({n}−{k}\:={p}\right) \\ $$$$=\sum_{{p}=\mathrm{1}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{n}−{p}} \:{n}\left({n}+\mathrm{1}\right)….\left({n}+{n}−{p}\right)}{\left({n}−{p}\right)!\:{t}^{{p}} \left(\mathrm{2}{i}\right)^{\mathrm{2}{n}−{p}} }\:+\frac{\mathrm{1}}{{n}!}\xi\left(\mathrm{1}\right) \\ $$$$=\sum_{{p}=\mathrm{1}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{n}−{p}} {n}\left({n}+\mathrm{1}\right)….\left(\mathrm{2}{n}−{p}\right)}{\left({n}−{p}\right)!\:\left({x}−{i}\right)^{{p}} \left(\mathrm{2}{i}\right)^{\mathrm{2}{n}−{p}} }\:+\frac{\mathrm{1}}{{n}!}\xi\left(\mathrm{1}\right)\:\Rightarrow \\ $$$${a}_{{k}} =\frac{\left(−\mathrm{1}\right)^{{n}−{k}} {n}\left({n}+\mathrm{1}\right)….\left(\mathrm{2}{n}−{k}\right)}{\left({n}−{k}\right)!\left(\mathrm{2}{i}\right)^{\mathrm{2}{n}−{k}} }\:\:\:\:{now}\:{let}\:{find}\:{b}_{{k}} \:\:{we}\:{use}\:{ch}\:.{x}+{i}\:={t}\:\Rightarrow \\ $$$${F}\left({x}\right)\:={H}\left({t}\right)\:=\frac{\mathrm{1}}{{t}^{{n}} \left({t}−\mathrm{2}{i}\right)^{{n}} }\:\:{let}\:{determine}\:{D}_{{n}−\mathrm{1}} \left(\mathrm{0}\right)\:{for}\:{v}\left({t}\right)\:=\left({t}−\mathrm{2}{i}\right)^{−{n}} \:\Rightarrow \\ $$$${v}\left({t}\right)\:=\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\:\frac{{v}^{\left({k}\right)} \left(\mathrm{0}\right)}{{k}!}\:{t}^{{k}} \:\:\:+{t}^{{n}} \:\xi\left({t}^{{n}} \right)\:\:\:\:{but}\:\:{v}^{\left({k}\right)} \left(\mathrm{0}\right)\:=\frac{\left(−\mathrm{1}\right)^{{k}} {n}\left({n}+\mathrm{1}\right)….\left({n}+{k}−\mathrm{1}\right)}{\left(−\mathrm{2}{i}\right)^{{n}+{k}} }\:\Rightarrow \\ $$$$\frac{{v}\left({t}\right)}{{t}^{{n}} }=\sum_{{p}=\mathrm{1}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{n}−{p}} {n}\left({n}+\mathrm{1}\right)…\left(\mathrm{2}{n}−{p}\right)}{\left({n}−{p}\right)!{t}^{{p}} \left(−\mathrm{2}{i}\right)^{\mathrm{2}{n}−{p}} }\:+\frac{\mathrm{1}}{{n}!}\xi\left(\mathrm{1}\right)\:\:\Rightarrow \\ $$$${b}_{{k}} =\frac{\left(−\mathrm{1}\right)^{{n}−{k}} {n}\left({n}+\mathrm{1}\right)….\left(\mathrm{2}{n}−{k}\right)}{\left({n}−{k}\right)!\left(−\mathrm{2}{i}\right)^{\mathrm{2}{n}−{k}} }\:\Rightarrow \\ $$$${F}\left({x}\right)\:=\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{n}−{k}} {n}\left({n}+\mathrm{1}\right)….\left(\mathrm{2}{n}−{k}\right)}{\left({n}−{k}\right)!\left(\mathrm{2}{i}\right)^{\mathrm{2}{n}−{k}} \left({x}−{i}\right)^{{k}} }\:+\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{n}−{k}} {n}\left({n}+\mathrm{1}\right)…\left(\mathrm{2}{n}−{k}\right)}{\left({n}−{k}\right)!\left(−\mathrm{2}{i}\right)^{\mathrm{2}{n}−{k}} \left({x}+{i}\right)^{{k}} } \\ $$$${and}\:{we}\:{see}\:{that}\:{b}_{{k}} ={conj}\left({a}_{{k}} \right)\:{so}\:{we}\:{can}\:{search}\:{only}\:{a}_{{k}} … \\ $$$${error}\:{of}\:{typing}\:\:{change}\:\xi\left({t}^{{n}} \right)\:{by}\:\xi\left({t}\right)… \\ $$
Commented by maxmathsup by imad last updated on 30/Apr/19
$${F}\left({x}\right)\:=\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{n}−{k}} \:{n}\left({n}+\mathrm{1}\right)…\left(\mathrm{2}{n}−{k}−\mathrm{1}\right)}{\left(\mathrm{2}{i}\right)^{} \:\:\left({n}−{k}\right)!\:\left({x}−{i}\right)^{{k}} }\:+\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{n}−{k}} {n}\left({n}+\mathrm{1}\right)…\left(\mathrm{2}{n}−{k}−\mathrm{1}\right)}{\left(−\mathrm{2}\boldsymbol{{i}}\right)^{\mathrm{2}\boldsymbol{{n}}−\boldsymbol{{k}}} \left(\boldsymbol{{n}}−\boldsymbol{{k}}\right)!\left({x}+{i}\right)^{{k}} } \\ $$