Hyperbolic Function - Inverse Functions As Logarithms

Inverse Functions As Logarithms

\begin{align} \operatorname {arsinh} (x) &= \ln \left(x + \sqrt{x^{2} + 1} \right) \\ \operatorname {arcosh} (x) &= \ln \left(x + \sqrt{x^{2} - 1} \right); x \ge 1 \\ \operatorname {artanh} (x) &= \frac{1}{2}\ln \left( \frac{1 + x}{1 - x} \right); \left| x \right| < 1 \\ \operatorname {arcoth} (x) &= \frac{1}{2}\ln \left( \frac{x + 1}{x - 1} \right); \left| x \right| > 1 \\ \operatorname {arsech} (x) &= \ln \left( \frac{1}{x} + \frac{\sqrt{1 - x^{2}}}{x} \right); 0 < x \le 1 \\ \operatorname {arcsch} (x) &= \ln \left( \frac{1}{x} + \frac{\sqrt{1 + x^{2}}}{\left| x \right|} \right); x \ne 0 \end{align}

Read more about this topic:  Hyperbolic Function

Famous quotes containing the words inverse and/or functions:

    “Yet time and space are but inverse measures of the force of the soul. The spirit sports with time.”
    —Ralph Waldo Emerson (1803–1882)

    “In today’s world parents find themselves at the mercy of a society which imposes pressures and priorities that allow neither time nor place for meaningful activities and relations between children and adults, which downgrade the role of parents and the functions of parenthood, and which prevent the parent from doing things he wants to do as a guide, friend, and companion to his children.”
    —Urie Bronfenbrenner (b. 1917)