Closed-form integral of w_j(t) over an interval

Computes \(\int_a^b w_j(t)\, dt\) in closed form using the inclusion-exclusion expansion. Each term integrates as: $$\int_a^b e^{-r t}\, dt = \frac{e^{-ra} - e^{-rb}}{r}$$

Usage

w_j_integral(a, b, par, j)

Arguments

a

Lower bound of integration (non-negative numeric scalar).

b

Upper bound of integration (positive numeric scalar, b > a).

par

Numeric vector of rates (length m).

j

Component index (integer, 1-based).

Value

Scalar integral value.

See also

w_j_exact() for the pointwise evaluation.

Examples

# Integral of w_1(t) from 0 to 1, rates = c(1, 2)
w_j_integral(0, 1, c(1, 2), j = 1)
#> [1] 0.3153829