Euler's formula
Euler's formula shows the coherence of the exponential function and hyperbolic functions
Explanation
The power series for the exponential function can be split into
Between parentheses 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.