Definition 17.1.

1.

hyperbolic sine of \(x\), written \(\sinh x\) is defined as: \[\sinh x=\frac {e^{\displaystyle {x}}-e^{\displaystyle {-x}}}{2},\quad x\in \mathbb {R}\]

2.

hyperbolic cosine of \(x\), written \(\cosh x\) is defined as: \[\cosh x =\frac {e^{\displaystyle {x}}+e^{\displaystyle {-x}}}{2},\quad x\in \mathbb {R}\]

3.

\[\tanh x =\frac {\sinh x}{\cosh x}=\frac {\frac {1}{2}(e^{\displaystyle {x}}-e^{\displaystyle {-x}})}{\frac {1}{2}(e^{\displaystyle {x}}+e^{\displaystyle {-x}})}=\frac {e^{\displaystyle {x}}-e^{\displaystyle {-x}}}{e^{\displaystyle {x}}+e^{\displaystyle {-x}}}\] \[\therefore \quad \tanh x =\frac {e^{\displaystyle {2x}}-1}{e^{\displaystyle {2x}}+1}, \quad x\in \mathbb {R}\]

4.

sech \(x=\frac {1}{\cosh x} =\frac {2}{\displaystyle {e^{\displaystyle {x}}+e^{\displaystyle {-x}}}},\quad x\in \mathbb {R}\)

5.

cosech\(x=\frac {1}{\sinh x}=\frac {2}{\displaystyle {e^{\displaystyle {x}}-e^{\displaystyle {-x}}}},\quad x\in \mathbb {R}, x\neq 0\)

6.

\(\coth x =\frac {\cosh x}{\sinh x}=\frac {\displaystyle {e^{\displaystyle {2x}}+1}}{\displaystyle {e^{\displaystyle {2x}}-1}},\quad x\in \mathbb {R},x\neq 0\)