Merge pull request #873 from Mathics3/add-ComplexExpand

Add complex expand

CHANGES.rst

@@ -104,6 +104,7 @@ New Builtins
 #. ``$PythonImplementation``
 #. ``Accuracy``
 #. ``ClebschGordan``
+#. ``ComplexExpand`` (@yzrun)
 #. ``Curl`` (2-D and 3-D vector forms only)
 #. ``DiscretePlot``
 #. ``Kurtosis``

SYMBOLS_MANIFEST.txt

@@ -226,6 +226,7 @@ System`CompiledCodeBox
 System`CompiledFunction
 System`Complement
 System`Complex
+System`ComplexExpand
 System`ComplexInfinity
 System`Complexes
 System`CompositeQ

mathics/builtin/numbers/hyperbolic.py

@@ -15,11 +15,14 @@
 from typing import Optional
 
 from mathics.builtin.arithmetic import _MPMathFunction
-from mathics.builtin.base import Builtin
+from mathics.builtin.base import Builtin, SympyFunction
 from mathics.core.atoms import IntegerM1
 from mathics.core.convert.sympy import SympyExpression
+from mathics.core.evaluation import Evaluation
 from mathics.core.expression import Expression
+from mathics.core.list import ListExpression
 from mathics.core.symbols import Symbol, SymbolPower
+from mathics.eval.hyperbolic import eval_ComplexExpand
 
 SymbolArcCosh = Symbol("ArcCosh")
 SymbolArcSinh = Symbol("ArcSinh")
@@ -246,9 +249,68 @@ class ArcTanh(_MPMathFunction):
     sympy_name = "atanh"
 
 
+class ComplexExpand(SympyFunction):
+    """
+        (<url>
+        :SymPy:
+        https://docs.sympy.org/latest/
+    modules/core.html#sympy.core.expr.Expr.expand</url>, <url>:WMA:
+        https://reference.wolfram.com/language/ref/ComplexExpand.html
+        </url>)
+
+        <dl>
+          <dt>'ComplexExpand[$expr$]'
+          <dd>expands $expr$ assuming that all variables are real.
+        </dl>
+
+        Note: we get equivalent, but different results from WMA:
+
+        >> ComplexExpand[3^(I x)]
+         = 3 ^ (-Im[x]) Re[3 ^ (I Re[x])] + I Im[3 ^ (I Re[x])] 3 ^ (-Im[x])
+
+        Assume that both and are real:
+        >> ComplexExpand[Sin[x + I y]]
+         = Cosh[y] Sin[x] + I Cos[x] Sinh[y]
+
+        Take $x$ to be complex:
+
+        >> ComplexExpand[Sin[x], x]
+         = Cosh[Im[x]] Sin[Re[x]] + I Cos[Re[x]] Sinh[Im[x]]
+
+        Polynomials:
+        >> ComplexExpand[Re[z^5 - 2 z^3 - z + 1], z]
+         = 1 + Re[z] ^ 5 - 2 Re[z] ^ 3 - Re[z] - 10 Im[z] ^ 2 Re[z] ^ 3 + 5 Im[z] ^ 4 Re[z] + 6 Im[z] ^ 2 Re[z]
+
+        Trigonometric and hyperbolic functions
+        >> ComplexExpand[Cos[x + I y] + Tanh[z], {z}]
+         = Cos[x] Cosh[y] - I Sin[x] Sinh[y] + Cosh[Re[z]] Sinh[Re[z]] / (Cos[Im[z]] ^ 2 + Sinh[Re[z]] ^ 2) + I Cos[Im[z]] Sin[Im[z]] / (Cos[Im[z]] ^ 2 + Sinh[Re[z]] ^ 2)
+
+        Exponential and logarithmic functions:
+        >> ComplexExpand[Abs[2^z Log[2 z]], z]
+         = Abs[I Arg[Re[z] + I Im[z]] + Log[4 Im[z] ^ 2 + 4 Re[z] ^ 2] / 2] 2 ^ Re[z]
+
+        Specify that a variable is take to be complex:
+        >> ComplexExpand[Re[2 z^3 - z + 1], z]
+         = 1 - Re[z] + 2 Re[z] ^ 3 - 6 Im[z] ^ 2 Re[z]
+    """
+
+    summary_text = "expand a complex expression of real variables"
+    sympy_name = "expand"
+
+    def eval(self, expr, evaluation: Evaluation):
+        "ComplexExpand[expr_]"
+        return eval_ComplexExpand(expr, ListExpression())
+
+    def eval_with_complex_vars(self, expr, vars, evaluation: Evaluation):
+        "ComplexExpand[expr_, vars__]"
+        return eval_ComplexExpand(expr, vars)
+
+
 class Cosh(_MPMathFunction):
     """
-    <url>:WMA link:https://reference.wolfram.com/language/ref/Cosh.html</url>
+    <url>
+    :WMA link:
+    https://reference.wolfram.com/language/ref/Cosh.html</url>
 
     <dl>
       <dt>'Cosh[$z$]'

mathics/eval/hyperbolic.py

@@ -0,0 +1,22 @@
+"""
+Mathics3 builtins from mathics.core.numbers.hyperbolic
+"""
+from sympy import Symbol as SympySymbol
+
+from mathics.core.convert.sympy import from_sympy
+
+
+def eval_ComplexExpand(expr, vars):
+    sympy_expr = expr.to_sympy()
+    if hasattr(vars, "elements"):
+        sympy_vars = {v.to_sympy() for v in vars.elements}
+    else:
+        sympy_vars = {vars.to_sympy()}
+    # All vars are assumed to be real
+    replaces = [
+        (fs, SympySymbol(fs.name, real=True))
+        for fs in sympy_expr.free_symbols
+        if fs not in sympy_vars
+    ]
+    sympy_expr = sympy_expr.subs(replaces)
+    return from_sympy(sympy_expr.expand(complex=True))

test/builtin/numbers/test_hyperbolic.py

@@ -23,3 +23,14 @@ def test_gudermannian():
         ("Gudermannian[z]", "2 ArcTan[Tanh[z / 2]]"),
     ):
         check_evaluation(str_expr, str_expected)
+
+
+def test_complexexpand():
+    for str_expr, str_expected in (
+        ("ComplexExpand[Sin[x + I y]]", "Cosh[y]*Sin[x] + I*Cos[x]*Sinh[y]"),
+        (
+            "ComplexExpand[3^(I x)]",
+            "3 ^ (-Im[x]) Re[3 ^ (I Re[x])] + I Im[3 ^ (I Re[x])] 3 ^ (-Im[x])",
+        ),
+    ):
+        check_evaluation(str_expr, str_expected)