What is the area bounded by the Szegő Curve?

Asked May 19 '17 at 06:03

Active May 19 '17 at 06:12

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I was reading about the zeros of the truncated exponential series and learned about the Szegő Curve. Naturally, I wanted to know the area bounded by the curve.

I read the equation of the curve to be: $$e^{(1-x)}\sqrt{x^2 +y^2} = 1$$ and calculated the area using Wolfram Alpha to be about $.731445$. Is a nice closed form known?

edited May 19 '17 at 06:12

J. M. ain't a mathematician

75,051

asked May 19 '17 at 06:03

zerosofthezeta

5,646

Your leftmost limit is $-W(e^{-1})$; I'm not sure if there's a closed form for the area, and Mathematica seems to not be able to find one. – J. M. ain't a mathematician May 19 '17 at 06:31
In polar coordinates the curve is $re^{1-r\cos\theta}=1$. I wonder if that helps? Probably not. – May 19 '17 at 06:37
The corresponding integral in @Rahul's formulation is $$\int_0^{\pi }\sec^2(\theta)W\left(-\frac{\cos(\theta )}{e}\right)^2 ,\mathrm d\theta$$; not very tractable either. – J. M. ain't a mathematician May 19 '17 at 06:44

What is the area bounded by the Szegő Curve?

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