Complex trigonometric and hyperbolic functions
In complex analysis, the familiar trigonometric and hyperbolic functions can be extended naturally from the real domain to the entire complex plane. Their classical series expansions retain validity for all complex numbers, and their connections to the exponential function offer a unifying perspective.
The complex forms of sine and cosine are defined through their Maclaurin series, which are valid for every z \in \mathbb{C}:
\begin{aligned} \sin(z) & = \sum_{k=0}^\infty \frac{(-1)^k z^{2k+1}}{(2k+1)!} \\ \cos(z) & = \sum_{k=0}^\infty \frac{(-1)^k z^{2k}}{(2k)!} \end{aligned}
These coincide with their familiar real forms and emphasize the analytic nature of the functions. Using Euler’s identities, they also admit concise exponential expressions:
\begin{aligned} \sin(z) & = \frac{e^{iz} - e^{-iz}}{2i} \\ \cos(z) & = \frac{e^{iz} + e^{-iz}}{2} \end{aligned}
Similarly, the hyperbolic sine and cosine are defined in direct analogy, using the classical exponential:
\begin{aligned} \sinh(z) & = \frac{e^{z} - e^{-z}}{2} \\ \cosh(z) & = \frac{e^{z} + e^{-z}}{2} \end{aligned}
Other trigonometric and hyperbolic functions, like the tangent, cotangent, secant, and their hyperbolic counterparts, are defined in terms of these primary building blocks. Their properties, identities, and singularities follow from the analytic structure inherited from the exponential function.
A noteworthy aspect is that all these functions are entire, meaning their Taylor expansions converge everywhere in \mathbb{C}. This is a direct consequence of their definitions as combinations of entire functions e^{z} and e^{iz}, bypassing any need for routine convergence tests based on series coefficients.
Identities that connect trigonometric and hyperbolic functions further illustrate their symmetry:
\begin{aligned} \sin(iz) & = i\sinh(z) \\ \cos(iz) & = \cosh(z) \\ i\sin(z) & = \sinh(iz) \\ \cos(z) & = \cosh(iz) \end{aligned}
Expressing the exponential directly in terms of hyperbolic or trigonometric functions with complex arguments brings out this connection:
e^{z} = \cosh(z) + \sinh(z) = \cos(iz) - i\sin(iz)
This perspective simplifies the analytic study of these functions, clarifying that their global properties are inherited from the exponential mapping on the complex plane.
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