• Hyperbolic sine. The hyperbolic sine function is given by

  \(\displaystyle \sinh(x) = \frac{e^x - e^{-x}}{2}\)

• Hyperbolic cosine. The hyperbolic cosine function is given by

  \(\displaystyle \cosh(x) = \frac{e^x + e^{-x}}{2}\)

• Hyperbolic tangent. The hyperbolic tangent function is given by

  \(\displaystyle \tanh(x) = \frac{\sinh(x)}{\cosh(x)}\)

• Inverse hyperbolic sine. The inverse hyperbolic sine function is given by

  \(\displaystyle \text{arsinh}(x) = \ln \left ( x + \sqrt{x^2 + 1} \right ) \)

• Inverse hyperbolic cosine. The inverse hyperbolic cosine function is given by

  \(\displaystyle \text{arcosh}(x) = \ln \left ( x + \sqrt{x^2 - 1} \right ) \)

  It is defined only for \(x \ge 1\).

• Inverse hyperbolic tangent. The inverse hyperbolic tangent function is given by

  \(\displaystyle \text{artanh}(x) = \tfrac{1}{2} \ln \left ( \frac{1 + x}{1 - x} \right ) \)

  It is defined only for \(|x| \le 1\).

• Standard logistic function. The stanard logistic function is given by

  \(\displaystyle \text{logistic}(x) = \frac{e^x}{1 + e^x} = \tfrac{1}{2} + \tfrac{1}{2} \tanh(\tfrac{x}{2}) \)

To do so, organize your program according to the following public API:

public class Hyperbolic {

    // Returns the hyperbolic sine of x.
    public static double sinh(double x)

    // Returns the hyperbolic cosine of x.
    public static double cosh(double x)

    // Returns the hyperbolic tangent of x.
    public static double tanh(double x)

    // Returns the inverse hyperbolic sine of x.
    public static double arsinh(double x)

    // Returns the inverse hyperbolic cosine of x.
    public static double arcosh(double x)

    // Returns the inverse hyperbolic tangent of x.
    public static double artanh(double x)

    // Returns the standard logistic function of x.
    public static double logistic(double x)

    // Takes a double command-line argument n and prints totient(n).
    public static void main(String[] args)
}

Your implementations should be one-liners, perhaps with extra code to handle corner cases.

• Undefined values. If a function is not defined for a given value of \( x \), return Double.NaN.

• Special values. By convention, \( \text{artanh}(1) = \infty\) and \( \text{artanh}(-1) = -\infty\).

Hyperbolic functions occur in the solutions of Laplace’s equation, which arise in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity. Logistic and hyperbolic tangent functions are widely used as activation functions in neural networks.