Question Number 124698 by Algoritm last updated on 05/Dec/20
Answered by Olaf last updated on 05/Dec/20
$$\mathrm{Let}\:{f}\left({x}\right)\:=\:\mathrm{ln}\left(\mathrm{3}+{x}^{\mathrm{2}} \right) \\ $$$${f}\left({x}\right)\:=\:\mathrm{ln}\mid{x}−\sqrt{\mathrm{3}}{i}\mid+\mathrm{ln}\mid{x}+\sqrt{\mathrm{3}}{i}\mid \\ $$$${f}'\left({x}\right)\:=\:\frac{\mathrm{1}}{{x}−\sqrt{\mathrm{3}}{i}}+\frac{\mathrm{1}}{{x}+\sqrt{\mathrm{3}}{i}} \\ $$$${f}''\left({x}\right)\:=\:−\frac{\mathrm{1}}{\left({x}−\sqrt{\mathrm{3}}{i}\right)^{\mathrm{2}} }−\frac{\mathrm{1}}{\left({x}+\sqrt{\mathrm{3}}{i}\right)^{\mathrm{2}} } \\ $$$${f}^{\left(\mathrm{3}\right)} \left({x}\right)\:=\:\frac{\mathrm{1}.\mathrm{2}}{\left({x}−\sqrt{\mathrm{3}}{i}\right)^{\mathrm{3}} }+\frac{\mathrm{1}.\mathrm{2}}{\left({x}+\sqrt{\mathrm{3}}{i}\right)^{\mathrm{3}} } \\ $$$${f}^{\left(\mathrm{4}\right)} \left({x}\right)\:=\:\left(−\mathrm{1}\right)^{\mathrm{3}} \left[\frac{\mathrm{1}.\mathrm{2}.\mathrm{3}}{\left({x}−\sqrt{\mathrm{3}}{i}\right)^{\mathrm{4}} }+\frac{\mathrm{1}.\mathrm{2}.\mathrm{3}}{\left({x}+\sqrt{\mathrm{3}}{i}\right)^{\mathrm{4}} }\right] \\ $$$$… \\ $$$${f}^{\left({n}\right)} \left({x}\right)\:=\:\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!\left[\frac{\mathrm{1}}{\left({x}−\sqrt{\mathrm{3}}{i}\right)^{{n}} }+\frac{\mathrm{1}}{\left({x}+\sqrt{\mathrm{3}}{i}\right)^{{n}} }\right] \\ $$$${x}\pm\sqrt{\mathrm{3}}{i}\:=\:\sqrt{{x}^{\mathrm{2}} +\mathrm{3}}.{e}^{\pm{i}\mathrm{arctan}\frac{\sqrt{\mathrm{3}}}{{x}}} \\ $$$${f}^{\left({n}\right)} \left({x}\right)\:=\:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{3}}}\left[\frac{\mathrm{1}}{\left({e}^{−{i}\mathrm{arctan}\frac{\sqrt{\mathrm{3}}}{{x}}} \right)^{{n}} }+\frac{\mathrm{1}}{\left({e}^{{i}\mathrm{arctan}\frac{\sqrt{\mathrm{3}}}{\:{x}}} \right)^{{n}} }\right] \\ $$$${f}^{\left({n}\right)} \left({x}\right)\:=\:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{3}}}\left[\frac{\mathrm{1}}{{e}^{−{in}\mathrm{arctan}\frac{\sqrt{\mathrm{3}}}{{x}}} }+\frac{\mathrm{1}}{{e}^{{in}\mathrm{arctan}\frac{\sqrt{\mathrm{3}}}{\:{x}}} }\right] \\ $$$${f}^{\left({n}\right)} \left({x}\right)\:=\:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{3}}}\left[{e}^{{in}\mathrm{arctan}\frac{\sqrt{\mathrm{3}}}{{x}}} +{e}^{−{in}\mathrm{arctan}\frac{\sqrt{\mathrm{3}}}{{x}}} \right] \\ $$$${f}^{\left({n}\right)} \left({x}\right)\:=\:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{3}}}\left[\mathrm{2cos}\left({n}\mathrm{arctan}\frac{\sqrt{\mathrm{3}}}{{x}}\right)\right] \\ $$$$\mathrm{or}\:\mathrm{we}\:\mathrm{can}\:\mathrm{write}\:: \\ $$$${f}^{\left({n}\right)} \left({x}\right)\:=\:\frac{\mathrm{2}\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{3}}}\mathrm{cos}\left[{n}\left(\frac{\pi}{\mathrm{2}}−\mathrm{arctan}\frac{{x}}{\:\sqrt{\mathrm{3}}}\right)\right] \\ $$
Answered by TANMAY PANACEA last updated on 05/Dec/20
$${y}={ln}\left(\mathrm{3}+{x}^{\mathrm{2}} \right) \\ $$$${y}_{\mathrm{1}} =\frac{\mathrm{2}{x}}{\mathrm{3}+{x}^{\mathrm{2}} }=\frac{\mathrm{2}{x}}{\left({x}+{i}\sqrt{\mathrm{3}}\:\right)\left({x}−{i}\sqrt{\mathrm{3}}\:\right)}=\frac{\mathrm{1}}{\left({x}+{i}\sqrt{\mathrm{3}}\:\right)}+\frac{\mathrm{1}}{\left({x}−{i}\sqrt{\left.\mathrm{3}\right)}\right.}=\frac{\mathrm{1}}{{x}+{a}}+\frac{\mathrm{1}}{{x}−{a}} \\ $$$${y}_{\mathrm{1}} =\left({x}+{a}\right)^{−\mathrm{1}} +\left({x}−{a}\right)^{−\mathrm{1}} \\ $$$${y}_{\mathrm{2}} =\left(−\mathrm{1}\right)\left({x}+{a}\right)^{−\mathrm{2}} +\left(−\mathrm{1}\right)\left({x}−{a}\right)^{−\mathrm{2}} \\ $$$${y}_{\mathrm{3}} =\left(−\mathrm{1}\right)\left(−\mathrm{2}\right)\left({x}+{a}\right)^{−\mathrm{3}} +\left(−\mathrm{1}\right)\left(−\mathrm{2}\right)\left({x}−{a}\right)^{−\mathrm{3}} \\ $$$$… \\ $$$$… \\ $$$${y}_{{n}} =\left(−\mathrm{1}\right)\left(−\mathrm{2}\right)…\left\{−\left({n}−\mathrm{1}\right)\right\}\left[\left({x}+{a}\right)^{−{n}} +\left({x}−{a}\right)^{−{n}} \right] \\ $$$${y}_{{n}} =\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!\left[\left({x}+{i}\sqrt{\mathrm{3}}\:\right)^{−{n}} +\left({x}−{i}\sqrt{\mathrm{3}}\:\right)^{−{n}} \right] \\ $$
Answered by Bird last updated on 05/Dec/20
$${f}\left({x}\right)={ln}\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)\:\Rightarrow \\ $$$${f}^{\left(\mathrm{1}\right)} \left({x}\right)=\frac{\mathrm{2}{x}}{{x}^{\mathrm{2}} \:+\mathrm{3}}=\frac{\mathrm{2}{x}}{\left({x}−{i}\sqrt{\mathrm{3}}\right)\left({x}+{i}\sqrt{\mathrm{3}}\right)} \\ $$$$=\frac{\mathrm{1}}{{x}−{i}\sqrt{\mathrm{3}}}+\frac{\mathrm{1}}{{x}+{i}\sqrt{\mathrm{3}}}\:\Rightarrow \\ $$$${f}^{\left({n}\right)} \left({x}\right)=\left(\frac{\mathrm{1}}{{x}−{i}\sqrt{\mathrm{3}}}\right)^{\left({n}−\mathrm{1}\right)} +\left(\frac{\mathrm{1}}{{x}+{i}\sqrt{\mathrm{3}}}\right)^{\left({n}−\mathrm{1}\right)} \\ $$$$=\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!}{\left({x}−{i}\sqrt{\mathrm{3}}\right)^{{n}} }+\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!}{\left({x}+{i}\sqrt{\mathrm{3}}\right)^{{n}} } \\ $$$$=\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!\left\{\frac{\mathrm{1}}{\left({x}−{i}\sqrt{\mathrm{3}}\right)^{{n}} }+\frac{\mathrm{1}}{\left({x}+{i}\sqrt{\mathrm{3}}\right)^{{n}} }\right\} \\ $$$$=\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!×\frac{\left({x}+{i}\sqrt{\mathrm{3}}\right)^{{n}} +\left({x}−{i}\sqrt{\mathrm{3}}\right)^{{n}} }{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{{n}} } \\ $$$${f}^{\left({n}\right)} \left({x}\right)=\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({n}−\mathrm{1}\right)!\frac{\mathrm{2}{Re}\left(\left({x}+{i}\sqrt{\mathrm{3}}\right)^{{n}} \right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{{n}} } \\ $$$${with}\:{n}\geqslant\mathrm{1} \\ $$$${we}\:{have}\:\left({x}+{i}\sqrt{\mathrm{3}}\right)^{{n}} \:=\sum_{{k}=\mathrm{0}} ^{{n}} {C}_{{n}} ^{{k}} \left({i}\sqrt{\mathrm{3}}\right)^{{k}} {x}^{{n}−{k}} \\ $$$$=\sum_{{p}=\mathrm{0}} ^{\left[\frac{{n}}{\mathrm{2}}\right]\:} {C}_{{n}} ^{\mathrm{2}{p}} \left(−\mathrm{1}\right)^{{p}} \:\mathrm{3}^{{p}} \:{x}^{{n}−\mathrm{2}{p}} \:+{is}\left({x}\right) \\ $$$$\Rightarrow{Re}\left(\left({x}+{i}\sqrt{\mathrm{3}}\right)^{{n}} \right) \\ $$$$=\sum_{{p}=\mathrm{0}} ^{\left[\frac{{n}}{\mathrm{2}}\sqrt{}\right.} \:\left(−\mathrm{1}\right)^{{p}} \:\mathrm{3}^{{p}} \:{C}_{{n}} ^{\mathrm{2}{p}\:} {x}^{{n}−\mathrm{2}{p}} \\ $$