I am interested in a function: $$\exp\left(-\frac{1}{1-x^2}\right)$$ on $(-1,1)$ and $0$ otherwise. According to Wikipedia this function is called bump function. In one variable it is a "bump". On the right side of the wikipedia page there is a plot of the bump function in two variables. It looks like croissant. I tried to plot $$\exp\left(-\frac{1}{1-x^2-y^2}\right)$$ in Wolfram alpha. It has circular symmetry. There is no croissant. What am I doing wrong?

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    $\begingroup$ The Wikipedia picture illustrates a bump function, not "the bump function" that you have plotted... $\endgroup$ – Hans Lundmark Aug 30 '17 at 10:11

The bump function $f(x,y)$, defined on $x^2+y^2 \leq 1$ $$ f(x,y) = \exp{\left(-\frac{1}{1-x^2-y^2}\right)} $$ and $f(x,y) = 0$ otherwise has radial symmetry.

To show it go in polar coordinates: $$ \begin{align} x &= r \cos(\theta) \\ y &= r \sin(\theta) \end{align} $$

and observe that $r^2=x^2+y^2$ is the equation of a circle. So you get $$ f(r) = \exp{\left(-\frac{1}{1-r^2}\right)} $$ so $f$ depends only by radius ($r$) and not angle.

The picture you refering to represents a bump function (it is smooth, compactly supported and not analytic) but it is not $f(x,y)$.


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