arctan and arccot are undefined at their poles

BREAKING-CHANGES.md

@@ -921,6 +921,22 @@ conjugate as `0 - 1i`, in decimals, on the one instance that every `i` is, in ev
 |---|---|---|
 | `Complex.ImaginaryOne.Conjugate`, after `(1 / Complex.ImaginaryOne).Evaled` with `DowncastingEnabled` off | `0 - 1i` | `-i` |
 
+### `arctan` and `arccot` are undefined at their poles
+
+**Silent: a simplification that cancels an `arctan` or an `arccot` carries `not 1 + x ^ 2 = 0`, where it carried
+nothing.** Both have poles at `i` and `-i`, where they evaluate to an infinity, as `ln(0)` does, and the tangent's
+condition excludes its own poles the same way. Their condition said they were defined everywhere, so
+`arctan(x) - arctan(x)` simplified to `0`, which has a value at `i`. An antiderivative that keeps an `arctan` can now
+list its poles beside the other points it has no value at, which says nothing new
+([#2256](https://github.com/asc-community/AngouriMath/issues/2256)).
+
+| Input | Was (2.6.0) | Now |
+|---|---|---|
+| `"arctan(x) - arctan(x)".ToEntity().Simplify()` | `0` | `0 provided not 1 + x ^ 2 = 0` |
+| `"arccot(x) - arccot(x)".ToEntity().Simplify()` | `0` | `0 provided not 1 + x ^ 2 = 0` |
+| `"arctan(x)".ToEntity().DomainCondition` | `True` | `not 1 + x ^ 2 = 0` |
+| `"x^2*coth(a + 2*ln(x))^2".ToEntity().Integrate("x").Simplify()` | `… + C + x ^ 3 / 3` | `… + C + x ^ 3 / 3 provided not x ^ 2 / e ^ (-a) + 1 = 0 and not x ^ 2 / (-e ^ (-a)) + 1 = 0` |
+
 ### `max` and `min` are defined where their arguments are real
 
 **Silent: `max(x, x)` and `min(x, x)` simplify to `x provided x in RR`, where they were `x`.** The complex numbers are

Sources/.editorconfig

@@ -235,6 +235,9 @@ file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed
 [AngouriMath/Core/Budgets/*.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/ArctangentPolesTest.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/Algebra/SolveTest/ModulusEquationTest.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/Continuous/Integration/IntegralPatterns.cs

@@ -1479,16 +1479,26 @@ private static bool IsAnEvenRootOfASymbol(Entity expr)
         /// and the second is the constant-numerator case above.
         /// </summary>
         /// <remarks>
+        /// <para>
         /// With <c>a</c> a symbol the constant-numerator case is a piecewise with an arm for
         /// <c>a = 0</c>, and on that arm the multiple of the derivative, which divided by
         /// <c>a</c>, is not the integrand's: there it is <c>(px + q)/(bx + c)</c>, which is
         /// <c>px/b + (q - pc/b) ln(bx + c)/b</c>. Timofeev's <c>(b1 + c1 x)/(a + 2b x + c x^2)</c>
         /// had no antiderivative, with the arm declined for the symbol in front.
+        /// </para>
+        /// <para>
+        /// A numerator that is a multiple of the derivative leaves no constant part, and the
+        /// arctangent it would have is left out rather than multiplied by zero, as the higher
+        /// powers leave it out: <c>0 * arctan(2x/2)</c> is <c>0</c> only where the arctangent has
+        /// a value, so <c>x/(x^2 + 1)</c> was answered with <c>provided not 1 + (2x/2)^2 = 0</c>
+        /// once the arctangent's poles were in its condition, beside a logarithm that has none
+        /// there either. https://github.com/asc-community/AngouriMath/issues/2256
+        /// </para>
         /// </remarks>
         private static Entity IntegrateLinearOverQuadratic(
             Entity p, Entity q, Entity a, Entity b, Entity c, Entity denominator, Entity.Variable x)
             => WithTheArmForAZeroLeadingCoefficient(
-                IntegrateRationalQuadratic(q - p * b / (2 * a), a, b, c, x),
+                TreeAnalyzer.IsZero(q - p * b / (2 * a)) ? Entity.Number.Integer.Zero : IntegrateRationalQuadratic(q - p * b / (2 * a), a, b, c, x),
                 a,
                 p / (2 * a) * AntiderivativeLog(denominator),
                 LinearOverPowerOfLinear(p, q, b, c, 1, x));

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

@@ -228,7 +228,11 @@ Set.Interval interval when IntervalArithmetic.MonotoneOn(interval, -1, 1, increa
 
         public partial record Arctanf
         {
-            private protected override Entity IntrinsicCondition => Boolean.True;
+            // arctan(z) = (i/2)(ln(1 - iz) - ln(1 + iz)) has poles where 1 + z^2 = 0, at z = i and z = -i, where it
+            // evaluates to an infinity, as ln(0) does: undefined there, as the tangent is where its cosine is 0.
+            // The condition said True, and arctan(x) - arctan(x) simplified to 0, which has a value at i.
+            // https://github.com/asc-community/AngouriMath/issues/2256
+            private protected override Entity IntrinsicCondition => !(1 + Argument.Pow(2)).EqualTo(0);
 
             /// <inheritdoc/>
             protected override Entity InnerSimplify(bool isExact) =>
@@ -254,7 +258,9 @@ Set.Interval interval when IntervalArithmetic.Monotone(interval, increasing: tru
 
         public partial record Arccotanf
         {
-            private protected override Entity IntrinsicCondition => Boolean.True;
+            // arccot(z) = arctan(1/z), so its poles are arctan's, z = i and z = -i. At 0 it is pi/2.
+            // https://github.com/asc-community/AngouriMath/issues/2256
+            private protected override Entity IntrinsicCondition => !(1 + Argument.Pow(2)).EqualTo(0);
 
             /// <inheritdoc/>
             protected override Entity InnerSimplify(bool isExact) =>

Sources/Tests/UnitTests/Calculus/AConstantExponentialBesideAFoldedLogarithmIntegralTest.cs

@@ -74,14 +74,17 @@ private static void DifferentiatesBack(string integrand, int longest)
         /// <summary>
         /// An even power of <c>x</c>, where the factorisation reads a symbol and splits the
         /// biquadratic over it into arms on its sign -- <c>coth(a + 2 ln(x))/x^2</c> is 3,577
-        /// characters in the symbol and 413 as the exponential, which is kept.
+        /// characters in the symbol and 413 as the exponential, which is kept. The bound was 1,300
+        /// until the arctangent's poles joined the points the answer lists as having no value:
+        /// <c>x^2 coth(a + 2 ln(x))^2</c> keeps two arctangents and went from 1,128 characters to
+        /// 1,382 with their two conditions. https://github.com/asc-community/AngouriMath/issues/2256
         /// </summary>
         [Theory]
         [InlineData("coth(a + 2*ln(x))/x^2")]
         [InlineData("coth(a + 2*ln(x))^2/x^2")]
         [InlineData("coth(a + 2*ln(x))^2")]
         [InlineData("x^2*coth(a + 2*ln(x))^2")]
         [InlineData("tanh(a + 2*ln(x))^2")]
-        public void AnEvenPowerOfX(string integrand) => DifferentiatesBack(integrand, 1300);
+        public void AnEvenPowerOfX(string integrand) => DifferentiatesBack(integrand, 1400);
     }
 }

Sources/Tests/UnitTests/Convenience/ArctangentPolesTest.cs

@@ -0,0 +1,51 @@
+//
+// 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.Extensions;
+using Xunit;
+
+namespace AngouriMath.Tests.Convenience
+{
+    /// <summary>
+    /// <c>arctan</c> and <c>arccot</c> are undefined at their poles, <c>i</c> and <c>-i</c>, where they evaluate to an
+    /// infinity, as <c>ln(0)</c> does and as the tangent is where its cosine is 0. Their condition said <c>True</c>, and
+    /// <c>arctan(x) - arctan(x)</c> simplified to <c>0</c>, which has a value at <c>i</c>.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/2256">#2256</a>
+    /// </summary>
+    [Trait("Area", "Convenience")]
+    public sealed class ArctangentPolesTest
+    {
+        [Theory]
+        [InlineData("arctan(x)", "i", false)]
+        [InlineData("arctan(x)", "-i", false)]
+        [InlineData("arctan(x)", "2", true)]
+        [InlineData("arctan(x)", "0", true)]
+        [InlineData("arctan(x)", "1/2 + 6/5 * i", true)]
+        [InlineData("arccot(x)", "i", false)]
+        [InlineData("arccot(x)", "-i", false)]
+        [InlineData("arccot(x)", "0", true)]
+        [InlineData("arccot(x)", "-3", true)]
+        public void ItIsDefinedOffItsPoles(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);
+            var value = expression.ToEntity().Substitute("x", at.ToEntity()).Evaled;
+            Assert.Equal(defined, value is Entity.Number number && number.IsFinite);
+        }
+
+        [Theory]
+        [InlineData("arctan(x) - arctan(x)")]
+        [InlineData("arccot(x) - arccot(x)")]
+        public void ASimplificationHasNoValueAtAPole(string expression)
+        {
+            var simplified = expression.ToEntity().Simplify();
+            Assert.True(simplified.Substitute("x", "i".ToEntity()).Evaled.IsNaN, $"{simplified} at i");
+            Assert.Equal(Entity.Number.Integer.Zero, simplified.Substitute("x", 2).Evaled);
+        }
+    }
+}