Hyperbolic Function - Hyperbolic Functions For Complex Numbers

Hyperbolic Functions For Complex Numbers

Since the exponential function can be defined for any complex argument, we can extend the definitions of the hyperbolic functions also to complex arguments. The functions sinh z and cosh z are then holomorphic.

Relationships to ordinary trigonometric functions are given by Euler's formula for complex numbers:

\begin{align} e^{i x} &= \cos x + i \;\sin x \\ e^{-i x} &= \cos x - i \;\sin x \end{align}

so:

\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}

Thus, hyperbolic functions are periodic with respect to the imaginary component, with period ( for hyperbolic tangent and cotangent).

Hyperbolic functions in the complex plane

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