pastas.rfunc.Hantush.get_tmax_approximation
===========================================

.. py:method:: pastas.rfunc.Hantush.get_tmax_approximation(p: pastas.typing.ArrayLike, cutoff: float | None = None) -> float

   
   Approximates the time (tmax) when the step response reaches a specified cutoff.

   This analytical approximation is derived by evaluating the tail of the
   impulse response integral. The derivation relies on the following steps:
   1.  The tail integral is mapped to dimensionless time x = t/a.
   2.  Deep in the tail, the b/t term in the exponent is assumed negligible,
       reducing the integral to the standard Exponential Integral, E1(x).
   3.  E1(x) is approximated by its leading asymptotic term: exp(-x) / x.
   4.  Equating this to the remaining area (1 - cutoff) yields an equation
       of the form x * exp(x) = z, which is classically solved using the
       Lambert W function: x = W(z).

   To prevent crashing the calculation when rho is large (causing k0 to
   underflow and z to overflow), the math is translated into log-space.
   The exponentially scaled Bessel function (k0e) and the Wright Omega
   function (the exact analytical solution to y + ln(y) = log_z)
   are used. This provides a globally continuous, unbreakable equivalent to
   lambertw(z).

   :param p: Response function parameters `[A, a, b]`.
   :type p: array_like
   :param cutoff: The fraction of the total step response area reached at tmax.
                  Must be strictly between 0 and 1. Defaults to `self.cutoff` if
                  `cutoff is None`.
   :type cutoff: float, optional

   :returns: The approximated tmax value.
   :rtype: float















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