LambertW over the reals is defined where its branch is real

Sources/.editorconfig

@@ -255,6 +255,9 @@ file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed
 [Tests/UnitTests/Convenience/MatrixFunctionsInAnExpressionTest.cs]
 file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
 
+[Tests/UnitTests/Convenience/LambertWOverTheRealsTest.cs]
+file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
+
 [Tests/UnitTests/Common/ExponentialOfALogarithmTest.cs]
 file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
 

Sources/AngouriMath/Functions/Evaluation/Evaluation.Continuous/Evaluation.Continuous.Arithmetics.Classes.cs

@@ -753,8 +753,14 @@ private static Entity Numerically(Complex n, int m)
         public partial record LambertWf
         {
             // Every whole branch, and every z but 0 off the principal branch, where W_k(z) goes to -oo as ln z does.
+            // Over the reals, where a branch is real: W_0 for z >= -1/e and W_(-1) for -1/e <= z < 0, both -1 at
+            // -1/e; no other branch is real anywhere. The condition was the complex one under either reading, so it
+            // held for W_0(-1), which is -0.318 + 1.337i. https://github.com/asc-community/AngouriMath/issues/2256
             private protected override Entity IntrinsicCondition =>
-                Branch.In(MathS.Sets.Z) & (Branch.Equalizes(Integer.Zero) | !Argument.Equalizes(Integer.Zero));
+                Codomain < Domain.Complex
+                ? Branch.Equalizes(Integer.Zero) & Argument >= -1 / MathS.e
+                  | Branch.Equalizes(Integer.Create(-1)) & Argument >= -1 / MathS.e & Argument < 0
+                : Branch.In(MathS.Sets.Z) & (Branch.Equalizes(Integer.Zero) | !Argument.Equalizes(Integer.Zero));
 
             /// <inheritdoc/>
             protected override Entity InnerSimplify(bool isExact) =>

Sources/Tests/UnitTests/Convenience/LambertWOverTheRealsTest.cs

@@ -0,0 +1,54 @@
+//
+// Copyright (c) 2019-2026 Angouri.
+// AngouriMath is licensed under MIT.
+// Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
+// Website: https://am.angouri.org.
+//
+
+using AngouriMath;
+using AngouriMath.Core;
+using AngouriMath.Extensions;
+using Xunit;
+
+namespace AngouriMath.Tests.Convenience
+{
+    /// <summary>
+    /// Over the reals the Lambert W function is defined where its branch is real: <c>W_0</c> for <c>z &gt;= -1/e</c> and
+    /// <c>W_(-1)</c> for <c>-1/e &lt;= z &lt; 0</c>. The condition was the complex one under either reading, so it held for
+    /// <c>W_0(-1)</c>, which is <c>-0.318 + 1.337i</c>.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/2256">#2256</a>
+    /// </summary>
+    [Trait("Area", "Convenience")]
+    public sealed class LambertWOverTheRealsTest
+    {
+        [Theory]
+        [InlineData("lambertw(0, x)", "-3", false)]
+        [InlineData("lambertw(0, x)", "-1", false)]
+        [InlineData("lambertw(0, x)", "-1/e", true)]
+        [InlineData("lambertw(0, x)", "-1/5", true)]
+        [InlineData("lambertw(0, x)", "0", true)]
+        [InlineData("lambertw(0, x)", "2", true)]
+        [InlineData("lambertw(-1, x)", "-1/e", true)]
+        [InlineData("lambertw(-1, x)", "-1/5", true)]
+        [InlineData("lambertw(-1, x)", "-1/2", false)]
+        [InlineData("lambertw(-1, x)", "0", false)]
+        [InlineData("lambertw(-1, x)", "1", false)]
+        [InlineData("lambertw(1, x)", "1", false)]
+        public void OverTheRealsItIsWhereItsBranchIsReal(string expression, string at, bool defined)
+        {
+            var condition = expression.ToEntity().DomainConditionIn(Domain.Real).Substitute("x", at.ToEntity()).Evaled;
+            Assert.Equal(defined ? Entity.Boolean.True : Entity.Boolean.False, condition);
+        }
+
+        /// <summary>Over the complex numbers every branch is defined but off the principal one at 0.</summary>
+        [Theory]
+        [InlineData("lambertw(0, x)", "-3", true)]
+        [InlineData("lambertw(-1, x)", "1", true)]
+        [InlineData("lambertw(-1, x)", "0", false)]
+        public void OverTheComplexNumbersItIsAsItWas(string expression, string at, bool defined)
+        {
+            var condition = expression.ToEntity().DomainCondition.Substitute("x", at.ToEntity()).Evaled;
+            Assert.Equal(defined ? Entity.Boolean.True : Entity.Boolean.False, condition);
+        }
+    }
+}