From 42d5d932b4c83beeadc2a7325d6229a8eddc0444 Mon Sep 17 00:00:00 2001
From: Alexander Voigt <ahgvoigt@gmail.com>
Date: Sun, 22 Mar 2026 12:27:36 +0100
Subject: [PATCH 1/5] adding failing test, see issue #40

---
 test/Dual.jl     | 14 ++++++++++++++
 test/runtests.jl |  1 +
 2 files changed, 15 insertions(+)
 create mode 100644 test/Dual.jl

diff --git a/test/Dual.jl b/test/Dual.jl
new file mode 100644
index 0000000..5805226
--- /dev/null
+++ b/test/Dual.jl
@@ -0,0 +1,14 @@
+if isdefined(Base, :get_extension)
+    import ForwardDiff
+
+    function li_dual(n::Integer, z::Complex)
+        z_dual = complex(ForwardDiff.Dual(real(z)), ForwardDiff.Dual(imag(z)))
+        PolyLog.li(n, z_dual)
+    end
+
+
+    @testset "Dual" begin
+        z = 0.5 + 0.8im
+        @test li_dual(2, z) == PolyLog.li(2, z)
+    end
+end
diff --git a/test/runtests.jl b/test/runtests.jl
index 3740ffd..b9d414f 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -9,6 +9,7 @@ include("TestPrecision.jl")
 include("DataReader.jl")
 include("DataTester.jl")
 include("Digamma.jl")
+include("Dual.jl")
 include("Eta.jl")
 include("Factorial.jl")
 include("Harmonic.jl")

From bb9df0460b155990c5edb15aa748e9f51134b391 Mon Sep 17 00:00:00 2001
From: Alexander Voigt <ahgvoigt@gmail.com>
Date: Sun, 22 Mar 2026 13:10:03 +0100
Subject: [PATCH 2/5] test for approximate equality

---
 test/Dual.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/test/Dual.jl b/test/Dual.jl
index 5805226..f7a92c2 100644
--- a/test/Dual.jl
+++ b/test/Dual.jl
@@ -9,6 +9,6 @@ if isdefined(Base, :get_extension)
 
     @testset "Dual" begin
         z = 0.5 + 0.8im
-        @test li_dual(2, z) == PolyLog.li(2, z)
+        @test li_dual(2, z) ≈ PolyLog.li(2, z) rtol=1e-15
     end
 end

From 4d82c05b5da27350b3f6b0c078b2a5c70199e55d Mon Sep 17 00:00:00 2001
From: Alexander Voigt <ahgvoigt@gmail.com>
Date: Sun, 22 Mar 2026 13:10:17 +0100
Subject: [PATCH 3/5] match complex types for _li2 and li2_approx

---
 src/Li2.jl | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/Li2.jl b/src/Li2.jl
index bba31b6..9c3a3e9 100644
--- a/src/Li2.jl
+++ b/src/Li2.jl
@@ -237,7 +237,7 @@ li2(z::Real) = li2(Complex(z))
 
 _li2(z::ComplexF16) = oftype(z, _li2(ComplexF32(z)))
 
-function _li2(z::Complex{T})::Complex{T} where T
+function _li2(z::Complex{T})::Complex{T} where {T<:Union{Float32, Float64}}
     rz, iz = reim(z)
 
     if iszero(iz)
@@ -270,7 +270,7 @@ function _li2(z::Complex{T})::Complex{T} where T
     end
 end
 
-function _li2(z::Complex{BigFloat})::Complex{BigFloat}
+function _li2(z::Complex{T})::Complex{T} where T
     rz, iz = reim(z)
 
     if iszero(iz)

From 36e3203cf5173b04924835734f1d026e641d2f8c Mon Sep 17 00:00:00 2001
From: Alexander Voigt <ahgvoigt@gmail.com>
Date: Sun, 22 Mar 2026 13:27:30 +0100
Subject: [PATCH 4/5] adding comments

---
 src/Li2.jl | 2 ++
 1 file changed, 2 insertions(+)

diff --git a/src/Li2.jl b/src/Li2.jl
index 9c3a3e9..908e553 100644
--- a/src/Li2.jl
+++ b/src/Li2.jl
@@ -237,6 +237,7 @@ li2(z::Real) = li2(Complex(z))
 
 _li2(z::ComplexF16) = oftype(z, _li2(ComplexF32(z)))
 
+# overload for complex types for which li2_approx uses pre-computed Bernoulli numbers
 function _li2(z::Complex{T})::Complex{T} where {T<:Union{Float32, Float64}}
     rz, iz = reim(z)
 
@@ -270,6 +271,7 @@ function _li2(z::Complex{T})::Complex{T} where {T<:Union{Float32, Float64}}
     end
 end
 
+# overload for generic complex types
 function _li2(z::Complex{T})::Complex{T} where T
     rz, iz = reim(z)
 

From a12822f0e5300c35f4f951d91e77e905a5add2f1 Mon Sep 17 00:00:00 2001
From: Alexander Voigt <ahgvoigt@gmail.com>
Date: Mon, 23 Mar 2026 07:26:28 +0100
Subject: [PATCH 5/5] extend test to more orders

---
 test/Dual.jl | 11 +++++------
 1 file changed, 5 insertions(+), 6 deletions(-)

diff --git a/test/Dual.jl b/test/Dual.jl
index f7a92c2..ab24ee4 100644
--- a/test/Dual.jl
+++ b/test/Dual.jl
@@ -1,14 +1,13 @@
 if isdefined(Base, :get_extension)
     import ForwardDiff
 
-    function li_dual(n::Integer, z::Complex)
-        z_dual = complex(ForwardDiff.Dual(real(z)), ForwardDiff.Dual(imag(z)))
-        PolyLog.li(n, z_dual)
-    end
-
+    to_dual(z::Complex) = Complex(ForwardDiff.Dual(real(z)), ForwardDiff.Dual(imag(z)))
 
     @testset "Dual" begin
         z = 0.5 + 0.8im
-        @test li_dual(2, z) ≈ PolyLog.li(2, z) rtol=1e-15
+
+        for n in -10:10
+            @test PolyLog.li(n, to_dual(z)) ≈ PolyLog.li(n, z) rtol=1e-14
+        end
     end
 end