Question Number 1351 by 123456 last updated on 24/Jul/15 | ||
$$\mathcal{W}\left\{{f}\left({x}\right)\right\}\left({t}\right)=\underset{\mathrm{0}} {\overset{\mathrm{1}/{t}} {\int}}{f}\left({x}\right)\mathrm{ln}\left({xt}\right){dx},{t}>\mathrm{0} \\ $$ $$\mathcal{W}\left\{{f}\left({x}\right)+{g}\left({x}\right)\right\}\left({t}\right)\overset{?} {=}\mathcal{W}\left\{{f}\left({x}\right)\right\}\left({t}\right)+\mathcal{W}\left\{{g}\left({x}\right)\right\}\left({t}\right) \\ $$ $$\mathcal{W}\left\{{cf}\left({x}\right)\right\}\left({t}\right)\overset{?} {=}{c}\mathcal{W}\left\{{f}\left({x}\right)\right\}\left({t}\right) \\ $$ $$\mathcal{W}\left\{\mathrm{1}\right\}\left({t}\right)=? \\ $$ $$\mathcal{W}\left\{{x}\right\}\left({t}\right)=? \\ $$ $$\mathcal{W}\left\{{x}^{{n}} \right\}\left({t}\right)=?,{n}\in\mathbb{N} \\ $$ $$\mathcal{W}\left\{{f}'\left({x}\right)\right\}\left({t}\right)=? \\ $$ | ||
Commented by112358 last updated on 25/Jul/15 | ||
$$\mathcal{W}\left\{{f}\left({x}\right)\right\}\left({t}\right)=\int_{\mathrm{0}} ^{\mathrm{1}/{t}} {f}\left({x}\right){ln}\left({xt}\right){dx}\:,\:{t}>\mathrm{0} \\ $$ $${I}'{m}\:{guessing}\:{one}\:{may}\:{write}\:\mathcal{W}\left\{{f}\left({x}\right)+{g}\left({x}\right)\right\}\left({t}\right)\:{as} \\ $$ $$\mathcal{W}\left\{{f}\left({x}\right)+{g}\left({x}\right)\right\}\left({t}\right)=\int_{\mathrm{0}} ^{\mathrm{1}/{t}} \left\{{f}\left({x}\right)+{g}\left({x}\right)\right\}{ln}\left({xt}\right){dx} \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\int_{\mathrm{0}} ^{\mathrm{1}/{t}} {f}\left({x}\right){ln}\left({xt}\right){dx}+\int_{\mathrm{0}} ^{\mathrm{1}/{t}} {g}\left({x}\right){ln}\left({xt}\right){dx} \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathcal{W}\left\{{f}\left({x}\right)\right\}\left({t}\right)+\mathcal{W}\left\{{g}\left({x}\right)\right\}\left({t}\right) \\ $$ $$\mathcal{W}\left\{{cf}\left({x}\right)\right\}\left({t}\right)=\int_{\mathrm{0}} ^{\:\mathrm{1}/{t}} \left\{{cf}\left({x}\right)\right\}{ln}\left({xt}\right){dx},\:{t}>\mathrm{0} \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={c}\int_{\mathrm{0}} ^{\:\mathrm{1}/{t}} {f}\left({x}\right){ln}\left({xt}\right){dx} \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={c}\mathcal{W}\left\{{f}\left({x}\right)\right\}\left({t}\right) \\ $$ $${W}\left\{\mathrm{1}\right\}\left({t}\right)=\int_{\mathrm{0}} ^{\:\mathrm{1}/{t}} \mathrm{1}×{ln}\left({xt}\right){dx} \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\underset{{p}\rightarrow\mathrm{0}} {\mathrm{lim}}\int_{{p}} ^{\mathrm{1}/{t}} \left({lnx}+{lnt}\right){dx} \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\underset{{p}\rightarrow\mathrm{0}} {\mathrm{lim}}\left({xlnx}−{x}+{xlnt}\right)\mid_{{p}} ^{\mathrm{1}/{t}} \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\underset{{p}\rightarrow\mathrm{0}} {\mathrm{lim}}\left\{{p}\left({lntp}−\mathrm{1}\right)+\frac{\mathrm{1}}{{t}}\right\} \\ $$ $${This}\:{limit}\:{is}\:{indefinite}\:{so}\: \\ $$ $$\mathcal{W}\left\{\mathrm{1}\right\}\left({t}\right)\:{does}\:{not}\:{exist}. \\ $$ $$ \\ $$ $$ \\ $$ | ||
Commented by112358 last updated on 25/Jul/15 | ||
$$\mathcal{W}\left\{{x}\right\}\left({t}\right)=\underset{\mathrm{0}} {\overset{\mathrm{1}/{t}} {\int}}{xlnxt}\:{dx} \\ $$ $$\mathcal{W}\left\{{x}^{{n}} \right\}\left({t}\right)=\underset{\mathrm{0}} {\overset{\mathrm{1}/{t}} {\int}}{x}^{{n}} {lnxt}\:{dx} \\ $$ $$\mathcal{W}\left\{{f}^{'} \left({x}\right)\right\}\left({t}\right)=\underset{\mathrm{0}} {\overset{\mathrm{1}/{t}} {\int}}{f}^{'} \left({x}\right){lnxt}\:{dx} \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={f}\left({x}\right){lnxt}\mid_{\mathrm{0}} ^{\mathrm{1}/{t}} −\underset{\mathrm{0}} {\overset{\mathrm{1}/{t}} {\int}}\frac{{f}\left({x}\right)}{{x}}{dx} \\ $$ $${These}\:{three}\:{integrals}\:{do}\:{no}\:{exist}. \\ $$ $$ \\ $$ $$ \\ $$ $$ \\ $$ | ||