Inverse Hyperbolic Tangent Calculator

Inverse hyperbolic tangent (artanh or tanh⁻¹) returns the value whose hyperbolic tangent equals the input.

1. Domain: (−1, 1) — strictly between −1 and 1
2. Range: all real numbers (−∞, +∞)
3. Formula: artanh(x) = ½ ln((1+x)/(1−x))
4. Odd function: artanh(−x) = −artanh(x)
5. Derivative: d/dx[artanh(x)] = 1/(1 − x²)

The calculations inside an inverse hyperbolic tangent (artanh) fundamentally map ratios confined firmly between boundary limits toward potentially infinite continuous spaces.

To compute correctly natively spanning across log equations, physicists leverage the absolute formula expression:

artanh(x) = (1/2) * ln((1 + x) / (1 − x))

The derivative showcases incredible alignment with rational mathematics: d/dx [artanh(x)] = 1 / (1 − x²) for any variable structurally constrained inside (−1, 1).

• Hyperbolic Tangent (tanh): Compresses wide exponential infinities rigidly within asymptotes at −1 and 1. Highly prevalent in neural network smoothing layers.
• Inverse Hyperbolic Tangent (artanh): Performs the geometric explosion trick—converting strict fractional bounds [−1, 1] into wildly sprawling infinities traversing all real ranges.

The function is intensely pivotal across a myriad of cutting edge physical technologies:

1. Fluid Dynamics: Computing precise timeline velocities for large falling objects accounting completely accurately for aerodynamic air resistance terminal velocity matrices.
2. Artificial Intelligence: Transforming and stretching normalized bounded activation layers back into continuous variables within backpropagation gradients.