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

Read more about this topic:  Hyperbolic Function

Famous quotes containing the words functions, complex and/or numbers:

    “Let us stop being afraid. Of our own thoughts, our own minds. Of madness, our own or others’. Stop being afraid of the mind itself, its astonishing functions and fandangos, its complications and simplifications, the wonderful operation of its machinery—more wonderful because it is not machinery at all or predictable.”
    —Kate Millett (b. 1934)

    “In the case of all other sciences, arts, skills, and crafts, everyone is convinced that a complex and laborious programme of learning and practice is necessary for competence. Yet when it comes to philosophy, there seems to be a currently prevailing prejudice to the effect that, although not everyone who has eyes and fingers, and is given leather and last, is at once in a position to make shoes, everyone nevertheless immediately understands how to philosophize.”
    —Georg Wilhelm Friedrich Hegel (1770–1831)

    “What culture lacks is the taste for anonymous, innumerable germination. Culture is smitten with counting and measuring; it feels out of place and uncomfortable with the innumerable; its efforts tend, on the contrary, to limit the numbers in all domains; it tries to count on its fingers.”
    —Jean Dubuffet (1901–1985)