Fitting a cotangent curve

Question

Hi,

I want to fit this curve to a cotangent function. I have cot(x/2) but how do I change the curve of the graph?

Thank you!

Answer 1

I agree with the comment from Claude Leibovici : This does not look like a cotangent.

If we draw $\ln(y(x))$ instead of $y(x)$ the function appears more linear for small $x$ and large $x$ with a transition between both.

We can approach the whole function as a sum of two partial functions : $$\ln(y)=(a_1+b_1x)\phi_1(x)+(a_2+b_2x)\phi_2(x) \tag 1$$ with $\phi_1(x)\simeq\begin{cases}1\quad\text{small}\quad x\\ 0\quad\text{large}\quad x \end{cases} \quad\text{and}\quad \phi_2(x)\simeq\begin{cases}0\quad\text{small}\quad x\\ 1\quad\text{large}\quad x\end{cases} $

The functions $\phi(x)$ can be built with functions such as arctan, arctanh, erf, etc.

$$\phi_1(x)=\frac12-\frac{1}{\pi}\tan^{-1}\big(p_1(x-c_1)\big) \quad\text{and}\quad \phi_2(x)=\frac12+\frac{1}{\pi}\tan^{-1}\big(p_2(x-c_2)\big) \tag 2$$ The transition area is characterized with the parameters $c$ (location) and $p$ (extend).

Or alternatively : $$\phi_1(x)=\frac12-\frac{1}{2}\tanh^{-1}\big(p_1(x-c_1)\big) \quad\text{and}\quad \phi_2(x)=\frac12+\frac{1}{2}\tanh^{-1}\big(p_2(x-c_2)\big)$$ $$\phi_1(x)=\frac12-\frac{1}{2}\text{Erf}\big(p_1(x-c_1)\big) \quad\text{and}\quad \phi_2(x)=\frac12+\frac{1}{2}\text{Erf}\big(p_2(x-c_2)\big)$$

For example, with rough data coming from the graph published in the question :

Red curve : from data.

Black dot curve : Computed from equations $(1)$ and $(2)$.

This is not the least mean square fit. This was drawn with $a_1=1.42\:;\:b_1=-0.55\:;\:p_1=30\:;\:c_1=1.6$ and $a_2=2\:;\:b_2=-2.2\:;\:p_2=15\:;\:c_2=1.65$ . These "guessed" values could be used as starting values for non-linear fitting (iterative calculus) which was not done.

Probably simpler mathematical models exist. The best should be to base the mathematical form of equation on a physical model.