# Integrable function [closed]

Suppose that $$a, b, c_1$$ and $$c_2$$ are real constant.

Is there the necessary and sufficient conditions of $$a ,b, c_1,c_2$$ for the following integration is integrable? i.e. $$\int_1^{\infty}\int_1^{\infty}\int_1^{\infty}\frac{1}{~x^{a}~y^{b}~(x+y)^{c_1}~(x+y+t)^{c_2}}~t^{-\frac{1}{2}}e^{-\frac{1}{t}} dx dy dt < \infty.$$

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# I know the necessary and sufficient conditions of $$a ,b, c_1$$ for

$$\int_1^{\infty}\int_1^{\infty}\frac{1}{~x^{a}~y^{b}~(x+y)^{c_1}}dx dy < \infty$$

is $$a+c_1>1, b+c_1>1$$ and $$a+b+c_1>2.$$

• Have you put any work into understanding the answer to your previous question, and adapting that answer in this new setting? Jan 21 '19 at 19:34
• @LeeMosher Thanks. I try to use the same idea of my previous question's answer. But I have not got the result. Jan 21 '19 at 19:42

The factor $$e^{-1/t}$$ is $$\asymp1$$ for $$t>1$$, and so, it may be dropped. So, the integral in question is finite iff $$I_j<\infty$$ for all $$j=1,\dots,6$$, where $$$$I_j:=\iiint\limits_{R_j}\frac{dx\,dy\,dt}{x^a\,y^b\,(x+y)^{c_1}\,(x+y+t)^{c_2}\,t^{1/2}},$$$$ \begin{align} R_1&:=\{(x,y,t)\colon 1
We have $$$$I_1\asymp\int_1^\infty\frac{dx}{x^a}\int_x^\infty\frac{dy}{y^{b+c_1}} \int_y^\infty\frac{dt}{t^{c_2+1/2}},$$$$ so that $$I_1<\infty$$ iff $$$$c_2+1/2>1,\quad b+c_1+c_2>3/2,\quad a+b+c_1+c_2>5/2.$$$$ Similarly, $$I_2<\infty$$ iff $$$$c_2+1/2>1,\quad a+c_1+c_2>3/2,\quad a+b+c_1+c_2>5/2.$$$$
The integrals $$I_3,\dots,I_6$$ are treated similarly. As the result, the integral in question is finite iff $$$$c_2+1/2>1,\quad a+c_1+c_2>3/2,\quad b+c_1+c_2>3/2,\quad a+b+c_1+c_2>5/2.$$$$