From 489369a93c68194b09d3ba132d3af4130a7c8251 Mon Sep 17 00:00:00 2001
From: Alexey Shiklomanov <alexey.shiklomanov@gmail.com>
Date: Fri, 2 Jun 2023 14:54:29 -0400
Subject: [PATCH 1/2] Derivative of exponential integral

General form:

$$ \frac{d}{dx} E_\nu(x) = -E_{\nu - 1}(x) $$

Special case:

$$ \frac{d}{dx} E_0(x) = -\frac{e^{-x}}{x} $$

Source: https://en.wikipedia.org/wiki/Exponential_integral#Derivatives
---
 src/rules.jl | 6 ++++++
 1 file changed, 6 insertions(+)

diff --git a/src/rules.jl b/src/rules.jl
index cbd4448..7b9b59e 100644
--- a/src/rules.jl
+++ b/src/rules.jl
@@ -181,6 +181,8 @@ _abs_deriv(x) = sign(x)
 @define_diffrule SpecialFunctions.ellipe(m) =
     :( (SpecialFunctions.ellipe($m) - SpecialFunctions.ellipk($m)) / (2 * $m) )
 
+@define_diffrule SpecialFunctions.expint(x)    = :( -exp(-$x) / $x )
+
 # TODO:
 #
 # eta
@@ -236,6 +238,10 @@ _abs_deriv(x) = sign(x)
 @define_diffrule SpecialFunctions.logbeta(a, b)   =
     :( SpecialFunctions.digamma($a) - SpecialFunctions.digamma($a + $b)  ), :(  SpecialFunctions.digamma($b) - SpecialFunctions.digamma($a + $b)  )
 
+# derivative wrt to `ν` is not implemented
+@define_diffrule SpecialFunctions.expint(ν, x)    = 
+    :NaN, :( -SpecialFunctions.expint($ν - 1, $x) )
+
 # derivative wrt to `s` is not implemented
 @define_diffrule SpecialFunctions.zeta(s, z)      =
     :NaN, :( - $s * SpecialFunctions.zeta($s + 1, $z)  )

From fed2f7db8990c4ed1f89abaafc265cdbc8e80ba7 Mon Sep 17 00:00:00 2001
From: Alexey Shiklomanov <alexey.shiklomanov@gmail.com>
Date: Fri, 2 Jun 2023 17:00:46 -0400
Subject: [PATCH 2/2] Bump SpecialFunctions compatibility to >=1.1

Exponential integral (`expint`) was only introduced in 1.0...

https://github.com/JuliaMath/SpecialFunctions.jl/releases/tag/v1.0.0

...but with spurious domain errors, which were fixed in 1.1 (at least
fixed-enough that this package's tests pass.)

https://github.com/JuliaMath/SpecialFunctions.jl/releases/tag/v1.1.0
---
 Project.toml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Project.toml b/Project.toml
index 4829f3a..b697659 100644
--- a/Project.toml
+++ b/Project.toml
@@ -13,7 +13,7 @@ SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b"
 IrrationalConstants = "0.1.1, 0.2"
 LogExpFunctions = "0.3.2"
 NaNMath = "0.3, 1"
-SpecialFunctions = "0.10, 1.0, 2"
+SpecialFunctions = "1.1, 2"
 julia = "1.3"
 
 [extras]