Ramanujan's continued fractions

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The Indian mathematician Srinivasa Ramanujan developed a number of interesting closed-form expressions for non-simple continued fractions. These are continued fractions without the restriction that the numerators be 1.

These include the almost integers

{1\over 1+{e^{-2\pi}\over 1+{e^{-4\pi}\over 1+\dots}}} = \left({\sqrt{5+\sqrt{5}\over 2}-{\sqrt{5}+1\over 2}}\right)e^{2\pi/5} = e^{2\pi/5}\left({\sqrt{\phi\sqrt{5}}-\phi}\right) = 0.9981360\dots

where φ is the golden ratio,

its multiplicative inverse

1 + {e^{-2\pi}\over 1+{e^{-4\pi}\over 1+{e^{-6\pi}\over 1+\dots}}}=\frac{1}{2}\left[1+\sqrt{5}+\sqrt{2(5+\sqrt{5})}\right]\,e^{-2\pi/5}
= \frac{e^{-2\pi/5}}{\sqrt{\phi\sqrt{5}}-{\phi}}=1.0018674.......

and

{1\over 1+{e^{-2\pi\sqrt{5}}\over 1+{e^{-4\pi\sqrt{5}}\over 1+\dots}}}
=\left(\frac{\sqrt{5}}{1+[5^{3/4}(\phi-1)^{5/2}-1]^{1/5}}-{\phi}\right)\,e^{2\pi/\sqrt{5}}=0.99999920....

And its multiplicative inverse,

1 + {e^{-2\pi\sqrt{5}}\over 1+{e^{-4\pi\sqrt{5}}\over 1+\dots}}
=\frac{e^{-2\pi/\sqrt{5}}}{\frac{\sqrt{5}}{1+\left[5^{3/4}(\phi-1)^{5/2}-1\right]^{1/5}}-{\phi}}=1.000000791267...

Many fascinating mathematical identities are expressed by continued fractions. One such is this: when

u={x\over 1}+\,{x^5\over 1}+\,{x^{10}\over 1}+\,{x^{15}\over 1} + \dots

and

v={x^{\frac{1}{5}}\over 1}+\,{x\over 1}+\,{x^2\over 1}+\,{x^3\over 1} + \dots

then

v^5=u\,\left[\frac{1-2u+4u^2-3u^3+u^4}{1+3u+4u^2-2u^3+u^4}\right].
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