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Hyperbolic sine

The hyperbolic sine can be expressed with exponential functions as



In the power series for the hyperbolic sine are only odd exponents and all terms have a plus sign. After duplication you get

To this we add even exponents and subtract these immediately

Reshuffling gives

In brackets are two power series for the exponential function and thus

so that


Example 1

You can see that sinh (0) = 0, because


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