In 1757, V. Riccati first recorded the generalizations of the hyperbolic functions defined by
(1)
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for , ..., , where is complex, with the value at defined by
(2)
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This is called the -hyperbolic function of order of the th kind. The functions satisfy
(3)
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where
(4)
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In addition,
(5)
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The functions give a generalized Euler formula
(6)
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Since there are th roots of , this gives a system of linear equations. Solving for gives
(7)
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where
(8)
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is a primitive root of unity.
The Laplace transform is
(9)
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The generalized hyperbolic function is also related to the Mittag-Leffler function by
(10)
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(11)
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The values and give the exponential and circular/hyperbolic functions (depending on the sign of ), respectively.
(12)
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(13)
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In particular
(14)
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(15)
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(16)
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For , the first few functions are
(17)
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(18)
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(19)
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(20)
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(21)
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(22)
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(23)
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(24)
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(25)
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(26)
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