Menu Close

let-give-D-R-2-0-0-and-from-R-let-C-1-x-y-D-0-lt-x-2-y-2-1-C-2-x-y-D-x-2-y-2-1-study-the-convergence-of-I-C-1-dxdy-x-2-y-2-and-J-C-2-




Question Number 30741 by abdo imad last updated on 25/Feb/18
let give D= R_+ ^2  −{(0,0)} and α from R let  C_1 ={(x,y)∈ D/0<x^2  +y^2 ≤1 }  C_2  ={(x,y) ∈D / x^2  +y^2 ≥1} study the convergence of  I= ∫∫_C_1     ((dxdy)/(((√(x^2  +y^2 )) )^α ))  and J=∫∫_C_2    ((dxdy)/(((√(x^2  +y^2  )) )^α )) .
$${let}\:{give}\:{D}=\:{R}_{+} ^{\mathrm{2}} \:−\left\{\left(\mathrm{0},\mathrm{0}\right)\right\}\:{and}\:\alpha\:{from}\:{R}\:{let} \\ $$$${C}_{\mathrm{1}} =\left\{\left({x},{y}\right)\in\:{D}/\mathrm{0}<{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \leqslant\mathrm{1}\:\right\} \\ $$$${C}_{\mathrm{2}} \:=\left\{\left({x},{y}\right)\:\in{D}\:/\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \geqslant\mathrm{1}\right\}\:{study}\:{the}\:{convergence}\:{of} \\ $$$${I}=\:\int\int_{{C}_{\mathrm{1}} } \:\:\:\frac{{dxdy}}{\left(\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }\:\right)^{\alpha} }\:\:{and}\:{J}=\int\int_{{C}_{\mathrm{2}} } \:\:\frac{{dxdy}}{\left(\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:}\:\right)^{\alpha} }\:. \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *