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
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}](../../../math/0/9/a/09a09deb4ccdc5ded033ad81d4d86e5f.png)
and
![=\left(\frac{\sqrt{5}}{1+[5^{3/4}(\phi-1)^{5/2}-1]^{1/5}}-{\phi}\right)\,e^{2\pi/\sqrt{5}}=0.99999920....](../../../math/1/2/6/1268098376a29abff28b5707a95d7023.png)
And its multiplicative inverse,
![=\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...](../../../math/4/2/0/42099e9ea7432102080737899478ac95.png)
Many fascinating mathematical identities are expressed by continued fractions. One such is this: when
and
then
![v^5=u\,\left[\frac{1-2u+4u^2-3u^3+u^4}{1+3u+4u^2-2u^3+u^4}\right].](../../../math/e/6/5/e6587c93f2edf6fdb8b947b0b9735795.png)
