All Questions

I am interested in the scaling of $$F(x_1,x_4)=\int_{\mathbb R^2} e^{-\vert x_1 -x_2 \vert -\varepsilon \vert x_2 -x_3 \vert- \vert x_3 -x_4 \vert } \ dx_2 dx_3 $$ In particular, I suspect that $$F(...

Is there a direct analytic proof that the function $$ f ( t ) = \exp\left(-|t|^\alpha \big[ \lambda + i \theta \operatorname{sign} ( t ) \big]\right), \qquad \lambda > 0, \quad |\theta| < \...