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Exponential function

The exponential function gives Euler's formula in the form

 


Explanation

The power series for the exponential function can be split into

Between brackets are the power series for the hyperbolic cosine and the power series for the hyperbolic sine. You can write this as it is because both series are absolutely convergent. We establish

Substitution of x = −x in the exponential function gives

We establish

 


Further calculations

Adding these two equations gives

and subtract them from each other

These are the definitions of the hyperbolic sine and the hyperbolic cosine.

 


History

The Swiss mathematician Leonhard Euler described this formula in 1748.


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