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Hyperbolic sine
The hyperbolic sine can be expressed with exponential functions as
Explanation
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