R Under development (unstable) (2025-11-06 r88990 ucrt) -- "Unsuffered Consequences"
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> ### == expint: Exponential Integral and Incomplete Gamma Function ==
> ###
> ### Tests for the incomplete gamma function
> ###
> ###    G(a,x) = int_x^infty t^{a-1} exp(-t) dt
> ###
> ### for a *real* and x >= 0.
> ###
> ### AUTHOR: Vincent Goulet <vincent.goulet@act.ulaval.ca>
> 
> ## Load the package
> library(expint)
> 
> ## a > 0; direct link to the standard incomplete gamma function
> x <- c(0.2, 2.5, 5, 8, 10)
> a <- 1.2
> stopifnot(exprs = {
+     identical(gammainc(a, x),
+               gamma(a) * pgamma(x, a, 1, lower = FALSE))
+ })
> 
> ## a = 0; direct link to the exponential integral
> x <- c(0.2, 2.5, 5, 8, 10)
> a <- 0
> stopifnot(exprs = {
+     identical(gammainc(a, x), expint(x))
+     identical(gammainc(a, x), expint_E1(x))
+ })
> 
> ## a < 0; compare to the recursive formula
> x <- c(0.2, 2.5, 5, 8, 10)
> a <- c(-0.25, -1.2, -2)
> stopifnot(exprs = {
+     all.equal(gammainc(a[1], x),
+               -(x^a[1] * exp(-x))/a[1] +
+               gamma(a[1] + 1) * pgamma(x, a[1] + 1, 1, lower = FALSE)/a[1])
+     all.equal(gammainc(a[2], x),
+               -(x^a[2] * exp(-x))/a[2] +
+               (-(x^(a[2] + 1) * exp(-x))/(a[2] + 1) +
+                gamma(a[2] + 2) * pgamma(x, a[2] + 2, 1, lower = FALSE)/(a[2] + 1))/a[2])
+     all.equal(
+         gammainc(a[3], x),
+         -(x^a[3] * exp(-x))/a[3] +
+         (-(x^(a[3] + 1) * exp(-x))/(a[3] + 1) + expint_E1(x)/(a[3] + 1))/a[3])
+ })
> 
> proc.time()
   user  system elapsed 
   0.18    0.07    0.25