Merge pull request #26471 from haru-44/LambertW_eval

Improvement of `LambertW` evaluation

sympy/functions/elementary/exponential.py

@@ -11,6 +11,7 @@
 from sympy.core.numbers import Integer, Rational, pi, I
 from sympy.core.parameters import global_parameters
 from sympy.core.power import Pow
+from sympy.core.relational import Ge
 from sympy.core.singleton import S
 from sympy.core.symbol import Wild, Dummy
 from sympy.core.sympify import sympify
@@ -1161,20 +1162,30 @@ def eval(cls, x, k=None):
                 return S.Zero
             if x is S.Exp1:
                 return S.One
-            if x == -1/S.Exp1:
-                return S.NegativeOne
+            w = Wild('w')
+            # W(x*log(x)) = log(x) for x >= 1/e
+            # e.g., W(-1/e) = -1, W(2*log(2)) = log(2)
+            result = x.match(w*log(w))
+            if result is not None and Ge(result[w]*S.Exp1, S.One) is S.true:
+                return log(result[w])
             if x == -log(2)/2:
                 return -log(2)
-            if x == 2*log(2):
-                return log(2)
+            # W(x**(x+1)*log(x)) = x*log(x) for x > 0
+            # e.g., W(81*log(3)) = 3*log(3)
+            result = x.match(w**(w+1)*log(w))
+            if result is not None and result[w].is_positive is True:
+                return result[w]*log(result[w])
+            # W(e**(1/n)/n) = 1/n
+            # e.g., W(sqrt(e)/2) = 1/2
+            result = x.match(S.Exp1**(1/w)/w)
+            if result is not None:
+                return 1 / result[w]
             if x == -pi/2:
                 return I*pi/2
             if x == exp(1 + S.Exp1):
                 return S.Exp1
             if x is S.Infinity:
                 return S.Infinity
-            if x.is_zero:
-                return S.Zero
 
         if fuzzy_not(k.is_zero):
             if x.is_zero:

sympy/functions/elementary/tests/test_exponential.py

@@ -605,7 +605,10 @@ def test_lambertw():
     assert LambertW(0) == 0
     assert LambertW(E) == 1
     assert LambertW(-1/E) == -1
+    assert LambertW(100*log(100)) == log(100)
     assert LambertW(-log(2)/2) == -log(2)
+    assert LambertW(81*log(3)) == 3*log(3)
+    assert LambertW(sqrt(E)/2) == S.Half
     assert LambertW(oo) is oo
     assert LambertW(0, 1) is -oo
     assert LambertW(0, 42) is -oo
@@ -627,6 +630,7 @@ def test_lambertw():
     assert LambertW(2, evaluate=False).is_real
     p = Symbol('p', positive=True)
     assert LambertW(p, evaluate=False).is_real
+    assert LambertW(p**(p+1)*log(p)) == p*log(p)
     assert LambertW(p - 1, evaluate=False).is_real is None
     assert LambertW(-p - 2/S.Exp1, evaluate=False).is_real is False
     assert LambertW(S.Half, -1, evaluate=False).is_real is False