Entry 9

Here is another "bizarre" continued fraction from Ramanujan involving \(e\) and \(\pi\). For \(x>0\) $$\sqrt{\frac{\pi\,e^x}{2x}}=1+\frac{x}{1\cdot3}+\frac{x^2}{1\cdot3\cdot5}+\frac{x^3}{1\cdot3\cdot5\cdot7}+\dots+\cfrac1{x+\cfrac{1}{1+\cfrac{2}{x+\cfrac{3}{1+\ddots}}}}$$As Kevin Brown of Mathpages commented in this old sci.math post, "Is there any other mathematician whose work is instantly recognizable?" Note that the error function has a reminiscent form (also rediscovered by Ramanujan) $$\int_0^x e^{-t^2}dt=\tfrac{1}{2}\sqrt{\pi}\,\text{erf}(x)=\tfrac{1}{2}\sqrt{\pi}-\cfrac1{x+\cfrac{1}{2x+\cfrac{2}{x+\cfrac{3}{2x+\ddots}}}}$$