Question Number 166735 by qaz last updated on 26/Feb/22
$$\mathrm{For}\:\:\mathrm{some}\:\mathrm{constant}\:\alpha\in\left(\mathrm{0},\mathrm{1}\right) \\ $$$$\mathrm{calculate}:\:\:\:\int_{\mathrm{0}} ^{\infty} \mathrm{e}^{−\frac{\mathrm{t}^{\alpha} }{\alpha}−\mathrm{xt}} \mathrm{dt}\sim\sqrt{\frac{\mathrm{2}\pi}{\mathrm{1}−\alpha}}\centerdot\mathrm{x}^{−\frac{\alpha}{\mathrm{2}\left(\mathrm{1}−\alpha\right)}−\mathrm{1}} \mathrm{e}^{\frac{\mathrm{1}−\alpha}{\alpha}\centerdot\mathrm{x}^{−\frac{\alpha}{\mathrm{1}−\alpha}} } ,\mathrm{x}\rightarrow\mathrm{0}^{+} \\ $$