Editing Hyperbolic functions (section)

==Hyperbolic functions for complex numbers==
{| style="text-align:center"
|+ Hyperbolic functions in the complex plane
|[[Image:Complex Sinh.jpg|1000x140px|none]]
|[[Image:Complex Cosh.jpg|1000x140px|none]]
|[[Image:Complex Tanh.jpg|1000x140px|none]]
|[[Image:Complex Coth.jpg|1000x140px|none]]
|[[Image:Complex Sech.jpg|1000x140px|none]]
|[[Image:Complex Csch.jpg|1000x140px|none]]
|-
|<math>\sinh(z)</math>
|<math>\cosh(z)</math>
|<math>\tanh(z)</math>
|<math>\coth(z)</math>
|<math>\operatorname{sech}(z)</math>
|<math>\operatorname{csch}(z)</math>
|}
Since the [[exponential function]] can be defined for any [[complex number|complex]] argument, we can also extend the definitions of the hyperbolic functions to complex arguments. The functions {{math|sinh ''z''}} and {{math|cosh ''z''}} are then [[Holomorphic function|holomorphic]].

Relationships to ordinary trigonometric functions are given by [[Euler's formula]] for complex numbers:
<math display="block">\begin{align}
  e^{i x} &= \cos x + i \sin x \\
 e^{-i x} &= \cos x - i \sin x
\end{align}</math>
so:
<math display="block">\begin{align}
   \cosh(ix) &= \frac{1}{2} \left(e^{i x} + e^{-i x}\right) = \cos x \\
   \sinh(ix) &= \frac{1}{2} \left(e^{i x} - e^{-i x}\right) = i \sin x \\
 \cosh(x+iy) &= \cosh(x) \cos(y) + i \sinh(x) \sin(y) \\
 \sinh(x+iy) &= \sinh(x) \cos(y) + i \cosh(x) \sin(y) \\
   \tanh(ix) &= i \tan x \\
     \cosh x &= \cos(ix) \\
     \sinh x &= - i \sin(ix) \\
     \tanh x &= - i \tan(ix)
\end{align}</math>

Thus, hyperbolic functions are [[periodic function|periodic]] with respect to the imaginary component, with period <math>2 \pi i</math> (<math>\pi i</math> for hyperbolic tangent and cotangent).