test real part of generalize li2(z)

Expander · Oct 6, 2023 · a9899dc · a9899dc
1 parent 92a5771
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diff --git a/src/Li2.jl b/src/Li2.jl
@@ -102,6 +102,26 @@ function li2_approx(u::ComplexF64)::ComplexF64
 end
 
 
+# series expansion of Li2(z) for |z| <= 1 and Re(z) <= 0.5
+# in terms of u = -log(1-z)
+function li2_approx(u::Complex{T})::Complex{T} where T
+    u2 = u*u
+    c0 = inv(4*big(pi)^2)
+    p = u2*c0^2
+    sum = one(T)/72
+
+    for n in 2:typemax(Int64)
+        old_sum = sum
+        sgn = iseven(n) ? -1 : 1
+        sum += sgn*p/(2*n + 1)*zeta(2*n, T)
+        sum == old_sum && break
+        p *= u2*c0
+    end
+
+    u + u2*(-one(T)/4 + 2*u*sum)
+end
+
+
 """
     reli2(x::Real)
 

diff --git a/test/Li2.jl b/test/Li2.jl
@@ -4,6 +4,8 @@
 
     setprecision(BigFloat, MAX_BINARY_DIGITS) do
         ep = 10*eps(BigFloat)
+        # @todo(alex): test imaginary part, too
+        test_function_on_data(z -> real(PolyLog.li2(z)), real_data, ep, ep)
         test_function_on_data(PolyLog.reli2, real_data, ep, ep)
     end