# Analytical solution for $\frac{x \cdot e^{-\frac{a^2}{8(1 + x^2)}} \sqrt{\frac{2}{\pi}}}{(1+x^2)^{1.5}} = \frac1{2 \sqrt{\pi} \cdot x^2}$?

Proof that solution exists is here. Let $$F(x) = \sqrt{\frac{2}{\pi}} \frac{x }{(1+x^2)^{3/2}} \exp\left(-\frac{a^2}{8(1 + x^2)} \right) - \frac1{2 \sqrt{\pi} \cdot x^2}$$ with $$a$$ being some fixed positive. Let calculate the limits:

1. $$\displaystyle\lim_{x \to 0} F(x) = -\infty$$
2. $$\displaystyle\lim_{x \to +\infty} F(x) = \lim_{x \to +\infty} \left(\frac{1 + O\left(\frac1{x^2}\right)}{x^2} - \frac1{x^2}\right) = \lim_{x \to +\infty} O\left(\frac1{x^4} \right) = +0$$.

Thus, $$F(x)$$ has one zero-crossing point at least.

I failed to find the analytical solution for $$F(x) = 0$$ with help of Wolfram Mathematica. Any ideas?

Solution in terms of the Lambert W function: $$x=\sqrt{\frac{12W\left(\frac{a^2}{24}\exp\left(\frac{a^2}{12}\right)\right)}{\left(a^2-12W\left(\frac{a^2}{24}\exp\left(\frac{a^2}{12}\right)\right)\right)} }$$
$$\sqrt{\frac{2}{\pi}} \frac{x }{(1+x^2)^{3/2}} \exp\left(-\frac{a^2}{8(1 + x^2)} \right) = \frac1{2 \sqrt{\pi} \cdot x^2} \\ 2^{3/2} \frac{x^3 }{(1+x^2)^{3/2}} \exp\left(-\frac{a^2}{8(1 + x^2)} \right) = 1$$ Both sides raised to power $$2/3$$ $$2 \frac{x^2 }{(1+x^2)} \exp\left(-\frac{a^2}{12(1 + x^2)} \right) = 1 \\ 2 \frac{x^2 }{(1+x^2)} \exp\left(\frac{a^2}{12}-\frac{a^2}{12(1 + x^2)} \right) = \exp\left(\frac{a^2}{12}\right) \\ 2 \frac{x^2 }{(1+x^2)} \exp\left(\frac{a^2x^2}{12(1+x^2)} \right) = \exp\left(\frac{a^2}{12}\right) \\ \frac{a^2x^2 }{12(1+x^2)} \exp\left(\frac{a^2x^2}{12(1+x^2)} \right) = \frac{a^2}{24}\exp\left(\frac{a^2}{12}\right) \\ \frac{a^2x^2 }{12(1+x^2)} = W\left(\frac{a^2}{24}\exp\left(\frac{a^2}{12}\right)\right) \\ a^2x^2 = 12(1+x^2)W\left(\frac{a^2}{24}\exp\left(\frac{a^2}{12}\right)\right) \\ x^2\left(a^2-12W\left(\frac{a^2}{24}\exp\left(\frac{a^2}{12}\right)\right)\right) = 12W\left(\frac{a^2}{24}\exp\left(\frac{a^2}{12}\right)\right) \\ x^2=\frac{12W\left(\frac{a^2}{24}\exp\left(\frac{a^2}{12}\right)\right)}{\left(a^2-12W\left(\frac{a^2}{24}\exp\left(\frac{a^2}{12}\right)\right)\right)} \\ x=\sqrt{\frac{12W\left(\frac{a^2}{24}\exp\left(\frac{a^2}{12}\right)\right)}{\left(a^2-12W\left(\frac{a^2}{24}\exp\left(\frac{a^2}{12}\right)\right)\right)} }$$