# Integral of $C^\infty$ Analog of Unit Step Function

The function $$f(x)\ :=\ e^{1-\frac{1}{\sqrt{1-x^2}}}$$ has the properties that

• $$\frac{d^nf}{dx^n}(\pm 1)=0$$,
• $$\frac{d^n}{dx^n}\left(\sqrt{1-x^2}-f(x)\right) = 0$$ at $$x=0$$,

and its integral $$F(x) := \int_{-1}^{x}f(t)\,\mathrm{d}t$$ provides $$C^\infty$$ transition between the two non-identical horizontal lines $$y(x)=0$$ and $$y(x)\approx1.48756$$.

Questions:

• can the integral $$F(x) := \int_{-1}^{x}e^{1-\frac{1}{\sqrt{1-t^2}}}\,\mathrm{d}t$$ be expressed by standard functions or known special functions?
• can at least $$F(1)$$ be expressed via already known functions?
• do $$f(x)$$ or $$F(x)$$ already appear as the solution to a mathematical or physical problem?

Remark: I "discovered" $$f(x)$$ when trying to find a $$C^\infty$$ connection between two parallel lines.

• Mathematica fails with both $F(x)$ and $F(1)$. – user64494 Oct 11 at 16:51
• Sorry for my ignorance, but what does the $\ast$ operation denote? – M.G. Oct 11 at 17:34
• What is $d^n\over x^n$? Is that meant to be $d^n\over dx^n$? – Gerry Myerson Oct 11 at 22:02
• I tried cleaning up the notation, and the typo Gerry pointed out. Please edit if I messed anything up. – David Roberts Oct 11 at 23:16
• In fact, if you just want a closed form smooth step function, it may be easier to look at things like $\tanh(\tan(x))$. – Willie Wong Oct 12 at 3:24