*3.3. DE Rules*

The double exponential transformation was first proposed by Takahasi and Mori in 1974 [16]. It can be seen from Equations (11)–(16) that the integration kernel is singular at *mi* = *ki*. DE rules are insensitive to endpoint singularity and simple to program since the weights and nodes are easily generated [12].

Consider the following form of integral:

$$I = \int\_{-1}^{1} f f(\xi) d\xi \tag{35}$$

Let a variable transform:

$$\zeta = \varrho(t) \text{ and } \varrho(-\infty) = -1, \varrho(+\infty) = 1 \tag{36}$$

be applied into (35) so as to change the interval [−1, 1] into the infinite interval [−∞, <sup>+</sup>∞]

$$I = \int\_{-\infty}^{+\infty} f f(\varphi(t)) \,\varphi'(t) dt\tag{37}$$

The DE rules are transformed by the tanh–sinh formula,

$$\varphi(t) = \tanh(\lg(t)) = \tanh(\sinh(t))\tag{38}$$

$$\text{g}'(t) = \text{g} \text{s}'(t) \text{sech}^2 \text{g} \text{s}(t) = \frac{\cosh(t)}{\cosh^2(\sinh(t))}\tag{39}$$

The standard trapezoidal rule for numerical integration is applied with *h* as grid interval when the integral is defined on the interval [−∞, <sup>+</sup>∞], and *n* is the sample point, which is truncated at ±*N*. Then we can approximate the definite integral via

$$I = h \sum\_{n = -\infty}^{\infty} f f\left(\varphi(nh)\right) \varphi'(nh) \approx h \sum\_{n = -N}^{N} \omega\_n f f\left(\xi\_n\right) \tag{40}$$

with the nodes *ξk* and weights *ωk* defined as

$$\mathcal{J}\_{\mathbb{H}} = 1 - \delta\_{\mathbb{H} \prime} \omega\_{\mathbb{H}} = 2 \operatorname{g}^{\prime}(nh) \delta\_{\mathbb{H}} \left( 1 - q\_{\mathbb{H}} \right)^{-1} \tag{41}$$

where

$$
\delta\_n = 2q\_n(1+q\_n)^{-1}, \\
q\_n = e^{-2\lg(nh)}\tag{42}
$$

For arbitrary integral interval [*a*, *b*] may be mapped onto [−1, 1] by the linear transformation *ξ* = *σx* + *γ* with *σ* = (*a* − *b*)/2, *γ* = (*a* + *b*)/2, which leads to

$$\int\_{-a}^{b} f f(\xi) d\xi = \sigma \int\_{-1}^{1} f f(\sigma \mathbf{x} + \gamma) d\mathbf{x} \tag{43}$$

Hence, for an arbitrary interval [*a*, *b*], the nodes and weights become *σξk* + *γ* and *σωk*, respectively, and (43) is transformed to

$$\int\_{-a}^{b} f f(\xi) d\xi \approx \sigma h \left\{ \text{g}s'(0) f f(\gamma) + \sum\_{n=1}^{N} \omega\_n [f f(a + \sigma \delta\_n) + f f(b - \sigma \delta\_n)] \right\} \tag{44}$$

As with any other quadrature rule, singularities of the integrand near the integration path adversely affect the convergence. However, any singularities on the integration path are easily treated by splitting the integration range so that the singularities are placed at the endpoints [12]. Therefore, the integration path in the first part of Equation (20) should be separated into (45) to ensure the convergence of the integration.

$$\int\_{-0}^{p} (\Gamma\_{\rm m} - \Gamma\_{\rm m}^{\infty}) f\_0(mr) dm = \left( \int\_{0}^{bk} + \int\_{\ \ \hbar k}^{p} \right) (\Gamma\_{\rm m} - \Gamma\_{\rm m}^{\infty}) f\_0(mr) dm \tag{45}$$

where breakpoint *bk* is set to the real part of wavenumber *ki* in this paper.
