If I've got $f: x \to f(x)$, one may define the Arithmetic Dissidence $\delta\_A[f(x)]$ as the real value of the difference between the length following the curve of $f$ and the length of the $x$ axis (we could define it on intervals or on the whole axis). You may also want to define the Geometric Dissidence $\delta\_G[f(x)]$ as the ratio of these lengths.


For example, if can be easily shown that for a function $f: x \to |x|$, $\delta\_G[f(x)]=\sqrt2$.  

For a distribution $\mathcal{T}(1-|x|)\_{[-1, 1]}$, $\delta\_G[\mathcal{T}]$ can't be different from $1$ but $\delta\_A[\mathcal{T}]=2(\sqrt2-1)$.


I need to find the value for $\delta\_A[f(x)]$ if it's finite for $f=\lambda e^{\frac{-x^2}{2\sigma^2}}$. Otherwise, I'd need the value of $\delta\_G[f(x)]$. I've still found no way of finding these values by integrating $d\ell=\sqrt{dx^2+df^2}$ but maybe you know one way.


Thanks for your help,



Getting a finite value for curve Dissidence
You seek the quantity
$$\delta\_G(x,\lambda)=\frac{1}{x}\int\_0^x \sqrt{1+f'(y)^2}\,dy,\;\;\text{with}\;\;f(x)=\lambda e^{-x^2/2},$$
where I have rescaled $x\mapsto x/\sigma$ to remove the parameter $\sigma$. There is no closed form answer for this integral, but for large $x$ it decays as
$$\delta\_G(x,\lambda)\approx 1+\lambda^2\frac{\sqrt{\pi}}{8x},\;\;x\gg 1,\;\;\lambda\ll 1.$$
For small $x$ it grows as
$$\delta\_G(x,\lambda)\approx 1+\lambda^2\frac{x^2}{6},\;\;x\ll 1.$$
Here is a plot of $\delta\_G(x,\lambda)$ for $\lambda=1$ (green) and the small-$x$ and large-$x$ asymptotes (blue and orange).


![](https://ilorentz.org/beenakker/MO/dissidence.png)


---


The alternative quantity
$$\delta\_A(x,\lambda)=\int\_0^x \sqrt{1+f'(y)^2}\,dy-x$$
approaches a constant in the limit $x\rightarrow\infty$. For $\lambda\ll 1$ I find
$$\lim\_{x\rightarrow\infty}\delta\_A(x,\lambda)=\frac{1}{8}\lambda^2\sqrt{\pi}.$$
Here is a plot of this large-$x$ limit as a function of $\lambda$. The orange curve is the exact result, the blue curve the small-$\lambda$ approximation.


![](https://ilorentz.org/beenakker/MO/dissidence_2.png)
