The polygamma functions, and the derivative of the factorial

BREAKING-CHANGES.md

@@ -78,6 +78,21 @@ as `a - b = 0`, as a statement's own solve reads it: `"x + 1 = 3".Solve("x")` wa
 | `MathS.Equations("x + y = 3", "x - y = 1").Solve("x", "y")` | `null` | `[[2, 1]]` |
 | `MathS.Equations("x + y = 3", "x - y = 1").ToString()` | `x + y = 3 = 0` and `x - y = 1 = 0` | `x + y - 3 = 0` and `x - y - 1 = 0` |
 
+### The polygamma functions, and the derivative of the factorial
+
+**Silent: a new reading, and an answer where there was none.**
+- **New functions.** `polygamma(n, z)` and `digamma(z)` are the polygamma functions `psi^(n)(z)`, the `(n + 1)`-th derivatives of `ln Gamma(z)`, and the digamma function, which is `polygamma(0, z)`. Neither name was recognised, so each was read as a product with an undeclared variable.
+- **Evaluation.** Each is evaluated to the working precision anywhere in the complex plane but at `0, -1, -2, ...`, its poles, where it is `NaN`.
+- **Derivatives.** Each order differentiates to the next. The derivative of `x!`, which was left unevaluated, is `x! * digamma(x + 1)` ([#171](https://github.com/asc-community/AngouriMath/issues/171)).
+
+| Input | Was (2.6.0) | Now |
+|---|---|---|
+| `"digamma(1)".ToEntity()` | `digamma * 1`, the variable `digamma` | `digamma(1)`, which evaluates to `-0.5772156649015328…` |
+| `"digamma(x)".ToEntity()` | `digamma * x` | `digamma(x)` |
+| `"polygamma(1, 1)".ToEntity()` | an `UnhandledParseException` | `polygamma(1, 1)`, which evaluates to `1.6449340668482264…` |
+| `"x!".ToEntity().Differentiate("x")` | `derivative(x!, x)`, unevaluated | `x! * digamma(x + 1)` |
+| `"gamma(x)".ToEntity().Differentiate("x")` | `derivative((x - 1)!, x)` | `(x - 1)! * digamma(x - 1 + 1)` |
+
 ### Numerical equality is to half the evaluation's digits
 
 **Silent: wrong answers.** `MathS.UnsafeAndInternal.AreEqualNumerically` evaluated both sides at its check points and

Sources/.editorconfig

@@ -696,6 +696,12 @@ file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed
 [AngouriMath/Core/Entity/Continuous/Number/ExponentialIntegrals.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
 
+[AngouriMath/Core/Entity/Continuous/Number/Polygamma.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
+
+[AngouriMath/Core/Entity/Continuous/Entity.Continuous.Polygamma.Classes.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
+
 [AngouriMath/Core/Entity/Continuous/Entity.Continuous.ExponentialIntegrals.Classes.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/Convenience/MathS.cs

@@ -1455,6 +1455,25 @@ public static Entity VectorLength(Matrix v)
         [MethodImpl(MethodImplOptions.AggressiveInlining)]
         public static Entity Li(Entity a) => new Lif(a);
 
+        /// <summary>The polygamma function, <c>psi^(n)(z)</c>, the <c>(n + 1)</c>-th derivative of <c>ln Gamma(z)</c></summary>
+        /// <param name="order">The order <c>n</c>, a whole number <c>n >= 0</c></param>
+        /// <param name="argument">The argument, any complex number but <c>0, -1, -2, ...</c>, its poles</param>
+        /// <returns>The <see cref="Entity.Polygammaf"/> node</returns>
+        /// <remarks>
+        /// <c>psi^(0)</c> is the digamma function, <c>Gamma'(z)/Gamma(z)</c>, which is what the derivative of
+        /// <c>x!</c> is written with: <c>(x!)' = x! psi(x + 1)</c>. Each order is the derivative of the one before.
+        /// https://github.com/asc-community/AngouriMath/issues/171
+        /// </remarks>
+        [MethodImpl(MethodImplOptions.AggressiveInlining)]
+        public static Entity Polygamma(Entity order, Entity argument) => new Polygammaf(order, argument);
+
+        /// <summary>The digamma function, <c>psi(z) = Gamma'(z)/Gamma(z)</c>, which is <c>polygamma(0, z)</c></summary>
+        /// <param name="argument">The argument, any complex number but <c>0, -1, -2, ...</c>, its poles</param>
+        /// <returns>The <see cref="Entity.Polygammaf"/> node of order <c>0</c></returns>
+        /// <remarks>https://github.com/asc-community/AngouriMath/issues/171</remarks>
+        [MethodImpl(MethodImplOptions.AggressiveInlining)]
+        public static Entity Digamma(Entity argument) => new Polygammaf(Integer.Zero, argument);
+
         /// <summary>The sine integral, <c>Si(z) = int_0^z sin(t)/t dt</c></summary>
         /// <param name="a">The argument, any complex number</param>
         /// <returns>The <see cref="Entity.Sif"/> node</returns>

Sources/AngouriMath/Core/Antlr/AngouriMath.g

@@ -616,6 +616,9 @@ atom returns[Entity value]
     /* The exponential and logarithmic integrals. https://github.com/asc-community/AngouriMath/issues/1501 */
     | 'Ei(' args = function_arguments ')' { Assert("Ei", 1, $args.list.Count); $value = MathS.Ei($args.list[0]); }
     | 'li(' args = function_arguments ')' { Assert("li", 1, $args.list.Count); $value = MathS.Li($args.list[0]); }
+    /* The polygamma functions, and the digamma function, which is the one of order 0. https://github.com/asc-community/AngouriMath/issues/171 */
+    | 'polygamma(' args = function_arguments ')' { Assert("polygamma", 2, $args.list.Count); $value = MathS.Polygamma($args.list[0], $args.list[1]); }
+    | 'digamma(' args = function_arguments ')' { Assert("digamma", 1, $args.list.Count); $value = MathS.Digamma($args.list[0]); }
     /* The trigonometric and hyperbolic integrals. https://github.com/asc-community/AngouriMath/issues/1501 */
     | 'Si(' args = function_arguments ')' { Assert("Si", 1, $args.list.Count); $value = MathS.Si($args.list[0]); }
     | 'Ci(' args = function_arguments ')' { Assert("Ci", 1, $args.list.Count); $value = MathS.Ci($args.list[0]); }

Sources/AngouriMath/Core/Antlr/AngouriMath.tokens

@@ -181,14 +181,16 @@ T__179=180
 T__180=181
 T__181=182
 T__182=183
-NEWLINE=184
-NUMBER=185
-SPECIALSET=186
-BOOLEAN=187
-NAN=188
-VARIABLE=189
-COMMENT=190
-WS=191
+T__183=184
+T__184=185
+NEWLINE=186
+NUMBER=187
+SPECIALSET=188
+BOOLEAN=189
+NAN=190
+VARIABLE=191
+COMMENT=192
+WS=193
 '!'=1
 '^'=2
 '-'=3
@@ -362,14 +364,16 @@ WS=191
 'erfi('=171
 'Ei('=172
 'li('=173
-'Si('=174
-'Ci('=175
-'Shi('=176
-'Chi('=177
-'trunc('=178
-'conjugate('=179
-'domain('=180
-'piecewise('=181
-'apply('=182
-'lambda('=183
-'NaN'=188
+'polygamma('=174
+'digamma('=175
+'Si('=176
+'Ci('=177
+'Shi('=178
+'Chi('=179
+'trunc('=180
+'conjugate('=181
+'domain('=182
+'piecewise('=183
+'apply('=184
+'lambda('=185
+'NaN'=190