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Michael Hardy
Michael Hardy
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Verify $ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\frac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}\frac{1}{\sqrt{1-\sqrt{x^2+y^2}}} dxdy\, dx \, dy <+\infty$

I want to know whether or not $$ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}\dfrac{1}{\sqrt{1-\sqrt{x^2+y^2}}} dxdy <+\infty.$$$$ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}} \frac{1}{\sqrt{1-\sqrt{x^2+y^2}}} \, dx \, dy <+\infty.$$

Here $D $ denotes the disk $\{(x,y)\in\mathbb{R}^2:x^2+y^2<1\}$.

Mathematica indicates that the limit is finite

In:= NIntegrate[(1/Sqrt[(x - 0.99999999999)^2 + 
         y^2])*(1/Sqrt[1 - Sqrt[x^2 + y^2]]), {x, -1, 1}, {y, -Sqrt[1 - x^2], Sqrt[1 - x^2]}]


Out:= 13.3727774796526

but along with the warning:

Numerical integration converging too slowly; suspect one of the 
following: singularity, value of the integration is 0, highly 
oscillatory integrand.

The $L^1$-norm of $\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}$ is uniformly bounded, and $\dfrac{1}{\sqrt{1-\sqrt{x^2+y^2}}}$ is integrable. However I don't know how to handle the singularity when the two are combined.

Verify $ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\frac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}\frac{1}{\sqrt{1-\sqrt{x^2+y^2}}} dxdy <+\infty$

I want to know whether or not $$ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}\dfrac{1}{\sqrt{1-\sqrt{x^2+y^2}}} dxdy <+\infty.$$

Here $D $ denotes the disk $\{(x,y)\in\mathbb{R}^2:x^2+y^2<1\}$.

Mathematica indicates that the limit is finite

In:= NIntegrate[(1/Sqrt[(x - 0.99999999999)^2 + 
         y^2])*(1/Sqrt[1 - Sqrt[x^2 + y^2]]), {x, -1, 1}, {y, -Sqrt[1 - x^2], Sqrt[1 - x^2]}]


Out:= 13.3727774796526

but along with the warning:

Numerical integration converging too slowly; suspect one of the 
following: singularity, value of the integration is 0, highly 
oscillatory integrand.

The $L^1$-norm of $\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}$ is uniformly bounded, and $\dfrac{1}{\sqrt{1-\sqrt{x^2+y^2}}}$ is integrable. However I don't know how to handle the singularity when the two are combined.

Verify $ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\frac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}\frac{1}{\sqrt{1-\sqrt{x^2+y^2}}} \, dx \, dy <+\infty$

I want to know whether or not $$ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}} \frac{1}{\sqrt{1-\sqrt{x^2+y^2}}} \, dx \, dy <+\infty.$$

Here $D $ denotes the disk $\{(x,y)\in\mathbb{R}^2:x^2+y^2<1\}$.

Mathematica indicates that the limit is finite

In:= NIntegrate[(1/Sqrt[(x - 0.99999999999)^2 + 
         y^2])*(1/Sqrt[1 - Sqrt[x^2 + y^2]]), {x, -1, 1}, {y, -Sqrt[1 - x^2], Sqrt[1 - x^2]}]


Out:= 13.3727774796526

but along with the warning:

Numerical integration converging too slowly; suspect one of the 
following: singularity, value of the integration is 0, highly 
oscillatory integrand.

The $L^1$-norm of $\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}$ is uniformly bounded, and $\dfrac{1}{\sqrt{1-\sqrt{x^2+y^2}}}$ is integrable. However I don't know how to handle the singularity when the two are combined.

No `\dfrac` in title; deleted "Thank you"
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LSpice
LSpice
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Verify $ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\dfrac\frac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}\dfrac\frac{1}{\sqrt{1-\sqrt{x^2+y^2}}} dxdy <+\infty$

I want to know whether or not $$ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}\dfrac{1}{\sqrt{1-\sqrt{x^2+y^2}}} dxdy <+\infty.$$

Here $D $ denotes the disk $\{(x,y)\in\mathbb{R}^2:x^2+y^2<1\}$.

Mathematica indicates that the limit is finite

In:= NIntegrate[(1/Sqrt[(x - 0.99999999999)^2 + 
         y^2])*(1/Sqrt[1 - Sqrt[x^2 + y^2]]), {x, -1, 1}, {y, -Sqrt[1 - x^2], Sqrt[1 - x^2]}]


Out:= 13.3727774796526

but along with the warning:

Numerical integration converging too slowly; suspect one of the 
following: singularity, value of the integration is 0, highly 
oscillatory integrand.

The $L^1$-norm of $\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}$ is uniformly bounded, and $\dfrac{1}{\sqrt{1-\sqrt{x^2+y^2}}}$ is integrable. However I don't know how to handle the singularity when the two are combined.

Thank for any suggestion.

Verify $ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}\dfrac{1}{\sqrt{1-\sqrt{x^2+y^2}}} dxdy <+\infty$

I want to know whether or not $$ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}\dfrac{1}{\sqrt{1-\sqrt{x^2+y^2}}} dxdy <+\infty.$$

Here $D $ denotes the disk $\{(x,y)\in\mathbb{R}^2:x^2+y^2<1\}$.

Mathematica indicates that the limit is finite

In:= NIntegrate[(1/Sqrt[(x - 0.99999999999)^2 + 
         y^2])*(1/Sqrt[1 - Sqrt[x^2 + y^2]]), {x, -1, 1}, {y, -Sqrt[1 - x^2], Sqrt[1 - x^2]}]


Out:= 13.3727774796526

but along with the warning:

Numerical integration converging too slowly; suspect one of the 
following: singularity, value of the integration is 0, highly 
oscillatory integrand.

The $L^1$-norm of $\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}$ is uniformly bounded, and $\dfrac{1}{\sqrt{1-\sqrt{x^2+y^2}}}$ is integrable. However I don't know how to handle the singularity when the two are combined.

Thank for any suggestion.

Verify $ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\frac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}\frac{1}{\sqrt{1-\sqrt{x^2+y^2}}} dxdy <+\infty$

I want to know whether or not $$ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}\dfrac{1}{\sqrt{1-\sqrt{x^2+y^2}}} dxdy <+\infty.$$

Here $D $ denotes the disk $\{(x,y)\in\mathbb{R}^2:x^2+y^2<1\}$.

Mathematica indicates that the limit is finite

In:= NIntegrate[(1/Sqrt[(x - 0.99999999999)^2 + 
         y^2])*(1/Sqrt[1 - Sqrt[x^2 + y^2]]), {x, -1, 1}, {y, -Sqrt[1 - x^2], Sqrt[1 - x^2]}]


Out:= 13.3727774796526

but along with the warning:

Numerical integration converging too slowly; suspect one of the 
following: singularity, value of the integration is 0, highly 
oscillatory integrand.

The $L^1$-norm of $\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}$ is uniformly bounded, and $\dfrac{1}{\sqrt{1-\sqrt{x^2+y^2}}}$ is integrable. However I don't know how to handle the singularity when the two are combined.

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Jessi
Jessi
  • 61
  • 3

Verify $ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}\dfrac{1}{\sqrt{1-\sqrt{x^2+y^2}}} dxdy <+\infty$

I want to know whether or not $$ \limsup_{\epsilon \rightarrow 0^+} \int_{D}\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}\dfrac{1}{\sqrt{1-\sqrt{x^2+y^2}}} dxdy <+\infty.$$

Here $D $ denotes the disk $\{(x,y)\in\mathbb{R}^2:x^2+y^2<1\}$.

Mathematica indicates that the limit is finite

In:= NIntegrate[(1/Sqrt[(x - 0.99999999999)^2 + 
         y^2])*(1/Sqrt[1 - Sqrt[x^2 + y^2]]), {x, -1, 1}, {y, -Sqrt[1 - x^2], Sqrt[1 - x^2]}]


Out:= 13.3727774796526

but along with the warning:

Numerical integration converging too slowly; suspect one of the 
following: singularity, value of the integration is 0, highly 
oscillatory integrand.

The $L^1$-norm of $\dfrac{1}{\sqrt{(x-(1-\epsilon))^2 +y^2}}$ is uniformly bounded, and $\dfrac{1}{\sqrt{1-\sqrt{x^2+y^2}}}$ is integrable. However I don't know how to handle the singularity when the two are combined.

Thank for any suggestion.