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You can find numerous equivalent methods for defining the hyperbolic capabilities. They might be outlined in terms of the exponential purpose:
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Considering that the exponential purpose might be outlined for any complex argument, we can increase the definitions with the hyperbolic features also to sophisticated arguments. The functions sinh z and cosh z are then holomorphic.
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The by-product of sinh x is cosh x plus the derivative of cosh x is sinh x; this is analogous to trigonometric capabilities, albeit the indicator is different (i.e., the derivative of cos x is −sin x).
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