Question Number 203199 by MathedUp last updated on 12/Jan/24
$$\mathrm{Let}'{s}\:\mathrm{define}\:\mathrm{linear}\:\mathrm{Operator}\:\boldsymbol{\mathcal{L}}\:\mathrm{as}\:\boldsymbol{\mathcal{L}}=\int_{\mathrm{0}} ^{\infty} \:{e}^{−{st}} \centerdot \\ $$$$\boldsymbol{\mathcal{L}}\left\{{W}\left({t}\right)\right\}=??? \\ $$$${W}\left({t}\right)\:\mathrm{is}\:\mathrm{inverse}\:\mathrm{function}\:\mathrm{of}\:{y}\left({t}\right)={te}^{{t}} \:,\:{t}\in\left[−\frac{\mathrm{1}}{{e}},\infty\right) \\ $$
Commented by shunmisaki007 last updated on 13/Jan/24
$$\mathrm{I}\:\mathrm{have}\:\mathrm{removed}\:\mathrm{my}\:\mathrm{answer}\:\mathrm{to}\:\mathrm{apologize}\: \\ $$$$\mathrm{my}\:\mathrm{misunderstanding}. \\ $$$$\mathrm{However},\:\mathrm{I}\:\mathrm{still}\:\mathrm{have}\:\mathrm{no}\:\mathrm{way}\:\mathrm{to}\:\mathrm{find} \\ $$$$\underset{\mathrm{0}} {\overset{\infty} {\int}}{e}^{−{st}} {W}\left({t}\right){dt}. \\ $$