Question Number 30741 by abdo imad last updated on 25/Feb/18
$${let}\:{give}\:{D}=\:{R}_{+} ^{\mathrm{2}} \:−\left\{\left(\mathrm{0},\mathrm{0}\right)\right\}\:{and}\:\alpha\:{from}\:{R}\:{let} \\ $$$${C}_{\mathrm{1}} =\left\{\left({x},{y}\right)\in\:{D}/\mathrm{0}<{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \leqslant\mathrm{1}\:\right\} \\ $$$${C}_{\mathrm{2}} \:=\left\{\left({x},{y}\right)\:\in{D}\:/\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \geqslant\mathrm{1}\right\}\:{study}\:{the}\:{convergence}\:{of} \\ $$$${I}=\:\int\int_{{C}_{\mathrm{1}} } \:\:\:\frac{{dxdy}}{\left(\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }\:\right)^{\alpha} }\:\:{and}\:{J}=\int\int_{{C}_{\mathrm{2}} } \:\:\frac{{dxdy}}{\left(\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:}\:\right)^{\alpha} }\:. \\ $$