Question Number 32739 by caravan msup abdo. last updated on 01/Apr/18
$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{e}^{−{t}} }{\mathrm{1}+{xt}}{dt} \\ $$$${calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right). \\ $$
Commented by abdo imad last updated on 03/Apr/18
![f is C^∞ and f^′ (x) =∫_0 ^∞ (∂/∂x)( (e^(−t) /(1+xt)))dt = ∫_0 ^∞ ((−t e^(−t) )/((1+xt)^2 )) dt also we have f(x) =∫_0 ^∞ (e^(−t) /t) (1/(x+(1/t)))dt ⇒ f^((n)) (x) = ∫_0 ^∞ (e^(−t) /t) (((−1)^n n!)/((x+(1/t))^(n+1) ))dt = (−1)^n n! ∫_0 ^∞ (e^(−t) /t) (t^(n+1) /((1+xt)^(n+1) )) dt =(−1)^n (n!) ∫_0 ^∞ ((t^n e^(−t) )/((1+xt)^(n+1) ))dt ⇒ f^((n)) (0) = (−1)^n (n!) ∫_0 ^∞ t^n e^(−t) dt let calculate A_(n ) =∫_0 ^∞ t^n e^(−t) dt .by parts A_n = [−t^n e^(−t) ]_0 ^∞ +∫_0 ^∞ n t^(n−1) e^(−t) dt =n A_(n−1) ⇒ Π_(k=1) ^n A_k =n! Π_(k=1) ^n A_(k−1) ⇒ A_n =n! A_0 =n! (look also that A_n =Γ(n+1) =n!) ⇒f^((n)) (0) =(−1)^n (n!)^2 .](Q32827.png)
$${f}\:{is}\:{C}^{\infty} \:\:{and}\:{f}^{'} \left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{\partial}{\partial{x}}\left(\:\frac{{e}^{−{t}} }{\mathrm{1}+{xt}}\right){dt} \\ $$$$=\:\int_{\mathrm{0}} ^{\infty} \:\frac{−{t}\:{e}^{−{t}} }{\left(\mathrm{1}+{xt}\right)^{\mathrm{2}} }\:{dt}\:\:{also}\:{we}\:{have}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−{t}} }{{t}}\:\frac{\mathrm{1}}{{x}+\frac{\mathrm{1}}{{t}}}{dt}\:\Rightarrow \\ $$$${f}^{\left({n}\right)} \left({x}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{t}} }{{t}}\:\frac{\left(−\mathrm{1}\right)^{{n}} \:{n}!}{\left({x}+\frac{\mathrm{1}}{{t}}\right)^{{n}+\mathrm{1}} }{dt} \\ $$$$=\:\left(−\mathrm{1}\right)^{{n}} \:{n}!\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{e}^{−{t}} }{{t}}\:\:\frac{{t}^{{n}+\mathrm{1}} }{\left(\mathrm{1}+{xt}\right)^{{n}+\mathrm{1}} }\:{dt} \\ $$$$=\left(−\mathrm{1}\right)^{{n}} \left({n}!\right)\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{n}} \:{e}^{−{t}} }{\left(\mathrm{1}+{xt}\right)^{{n}+\mathrm{1}} }{dt}\:\Rightarrow \\ $$$${f}^{\left({n}\right)} \left(\mathrm{0}\right)\:=\:\left(−\mathrm{1}\right)^{{n}} \left({n}!\right)\:\int_{\mathrm{0}} ^{\infty} \:{t}^{{n}} \:{e}^{−{t}} {dt}\:{let}\:{calculate} \\ $$$${A}_{{n}\:} =\int_{\mathrm{0}} ^{\infty} \:{t}^{{n}} \:{e}^{−{t}} {dt}\:.{by}\:{parts} \\ $$$${A}_{{n}} \:\:=\:\left[−{t}^{{n}} \:{e}^{−{t}} \right]_{\mathrm{0}} ^{\infty} \:\:+\int_{\mathrm{0}} ^{\infty} \:{n}\:{t}^{{n}−\mathrm{1}} \:{e}^{−{t}} {dt}\:={n}\:{A}_{{n}−\mathrm{1}} \:\Rightarrow \\ $$$$\prod_{{k}=\mathrm{1}} ^{{n}} \:{A}_{{k}} ={n}!\:\prod_{{k}=\mathrm{1}} ^{{n}} \:{A}_{{k}−\mathrm{1}} \:\Rightarrow\:{A}_{{n}} ={n}!\:{A}_{\mathrm{0}} ={n}!\:\left({look}\:{also}\right. \\ $$$$\left.{that}\:{A}_{{n}} =\Gamma\left({n}+\mathrm{1}\right)\:={n}!\right)\:\Rightarrow{f}^{\left({n}\right)} \left(\mathrm{0}\right)\:=\left(−\mathrm{1}\right)^{{n}} \left({n}!\right)^{\mathrm{2}} \:. \\ $$