By analogy with the lemniscate functions, hyperbolic lemniscate functions can also be defined
(1)
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(2)
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(3)
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(4)
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where is a hypergeometric function.
Let and , and write
(5)
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(6)
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where is the constant obtained by setting and , which is given by
(7)
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(8)
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with is a complete elliptic integral of the first kind. Ramanujan showed that
(9)
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(10)
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and
(11)
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(Berndt 1994).