Here is another "bizarre" continued fraction from Ramanujan involving
e and
π. For
x>0 √πex2x=1+x1⋅3+x21⋅3⋅5+x31⋅3⋅5⋅7+⋯+1x+11+2x+31+⋱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)
∫x0e−t2dt=12√πerf(x)=12√π−1x+12x+2x+32x+⋱
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