doctest changes and a few fixes in support of pynac-0.6.5

src/sage/functions/log.py

@@ -43,7 +43,7 @@ def __init__(self):
             sage: exp(RDF('2.5'))
             12.182493960703473
             sage: exp(I*pi/12)
-            1/12*sqrt(6)*(sqrt(3) + 3) - 1/12*I*sqrt(6)*(sqrt(3) - 3)
+            (1/4*I + 1/4)*sqrt(6) - (1/4*I - 1/4)*sqrt(2)
 
         To prevent automatic evaluation, use the ``hold`` parameter::
 

src/sage/functions/orthogonal_polys.py

@@ -317,7 +317,7 @@
 
 
 from sage.symbolic.ring import SR, is_SymbolicVariable
-from sage.symbolic.function import BuiltinFunction
+from sage.symbolic.function import BuiltinFunction, GinacFunction
 from sage.symbolic.expression import Expression
 from sage.functions.other import factorial, binomial
 from sage.structure.all import parent
@@ -1238,7 +1238,7 @@ def gen_legendre_Q(n, m, x):
         return ((n-m+1)*x*gen_legendre_Q(n,m-1,x)-(n+m-1)*gen_legendre_Q(n-1,m-1,x))/sqrt(1-x**2)
 
 
-def hermite(n, x):
+class Func_hermite(GinacFunction):
     """
     Returns the Hermite polynomial for integers `n > -1`.
 
@@ -1257,13 +1257,13 @@ def hermite(n, x):
         40
         sage: S.<y> = PolynomialRing(RR)
         sage: hermite(3,y)
-        8.00000000000000*y^3 - 12.0000000000000*y
+        8*y^3 - 12*y
         sage: R.<x,y> = QQ[]
         sage: hermite(3,y^2)
         8*y^6 - 12*y^2
         sage: w = var('w')
         sage: hermite(3,2*w)
-        8*(8*w^2 - 3)*w
+        64*w^3 - 24*w
 
     Check that :trac:`17192` is fixed::
 
@@ -1274,19 +1274,27 @@ def hermite(n, x):
         sage: hermite(-1,x)
         Traceback (most recent call last):
         ...
-        ValueError: n must be greater than -1, got n = -1
+        RuntimeError: hermite_eval: The index n must be a nonnegative integer
 
         sage: hermite(-7,x)
         Traceback (most recent call last):
         ...
-        ValueError: n must be greater than -1, got n = -7
+        RuntimeError: hermite_eval: The index n must be a nonnegative integer
     """
-    if not (n > -1):
-        raise ValueError("n must be greater than -1, got n = {0}".format(n))
+    def __init__(self):
+        r"""
+        Init method for the Hermite polynomials.
 
-    _init()
-    return sage_eval(maxima.eval('hermite(%s,x)'%ZZ(n)), locals={'x':x})
+        EXAMPLES::
+
+            sage: loads(dumps(hermite))
+            hermite
+        """
+        GinacFunction.__init__(self, "hermite", nargs=2, latex_name=r"H",
+                conversions={'maxima':'hermite', 'mathematica':'HermiteH',
+                    'maple':'HermiteH'})
 
+hermite = Func_hermite()
 
 def jacobi_P(n, a, b, x):
     r"""
@@ -1381,7 +1389,7 @@ def legendre_Q(n, x):
     return sage_eval(maxima.eval('legendre_q(%s,x)'%ZZ(n)), locals={'x':x})
 
 
-def ultraspherical(n, a, x):
+class Func_ultraspherical(GinacFunction):
     """
     Returns the ultraspherical (or Gegenbauer) polynomial for integers
     `n > -1`.
@@ -1394,6 +1402,8 @@ def ultraspherical(n, a, x):
 
     EXAMPLES::
 
+        sage: ultraspherical(8, 101/11, x)
+        795972057547264/214358881*x^8 - 62604543852032/19487171*x^6...
         sage: x = PolynomialRing(QQ, 'x').gen()
         sage: ultraspherical(2,3/2,x)
         15/2*x^2 - 3/2
@@ -1404,6 +1414,12 @@ def ultraspherical(n, a, x):
         sage: t = PolynomialRing(RationalField(),"t").gen()
         sage: gegenbauer(3,2,t)
         32*t^3 - 12*t
+        sage: var('x')
+        x
+        sage: for N in range(100):
+        ....:     n = ZZ.random_element().abs() + 5
+        ....:     a = QQ.random_element().abs() + 5
+        ....:     assert ((n+1)*ultraspherical(n+1,a,x) - 2*x*(n+a)*ultraspherical(n,a,x) + (n+2*a-1)*ultraspherical(n-1,a,x)).expand().is_zero()
 
     Check that :trac:`17192` is fixed::
 
@@ -1414,20 +1430,28 @@ def ultraspherical(n, a, x):
         sage: ultraspherical(-1,1,x)
         Traceback (most recent call last):
         ...
-        ValueError: n must be greater than -1, got n = -1
+        RuntimeError: gegenb_eval: The index n must be a nonnegative integer
 
         sage: ultraspherical(-7,1,x)
         Traceback (most recent call last):
         ...
-        ValueError: n must be greater than -1, got n = -7
+        RuntimeError: gegenb_eval: The index n must be a nonnegative integer
     """
-    if not (n > -1):
-        raise ValueError("n must be greater than -1, got n = {0}".format(n))
+    def __init__(self):
+        r"""
+        Init method for the ultraspherical polynomials.
 
-    _init()
-    return sage_eval(maxima.eval('ultraspherical(%s,%s,x)'%(ZZ(n),a)), locals={'x':x})
+        EXAMPLES::
+
+            sage: loads(dumps(ultraspherical))
+            gegenbauer
+        """
+        GinacFunction.__init__(self, "gegenbauer", nargs=3, latex_name=r"C",
+                conversions={'maxima':'ultraspherical', 'mathematica':'GegenbauerC',
+                    'maple':'GegenbauerC'})
 
-gegenbauer = ultraspherical
+ultraspherical = Func_ultraspherical()
+gegenbauer = Func_ultraspherical()
 
 
 class Func_laguerre(OrthogonalFunction):

src/sage/functions/trig.py

@@ -47,35 +47,54 @@ def __init__(self):
             1/4*sqrt(-2*sqrt(6) - 2*sqrt(2) + 8)
             sage: sin(pi/30)
             -1/8*sqrt(5) + 1/4*sqrt(-3/2*sqrt(5) + 15/2) - 1/8
+            sage: sin(104*pi/105)
+            sin(1/105*pi)
             sage: cos(pi/8)
             1/2*sqrt(sqrt(2) + 2)
             sage: cos(pi/10)
-            1/2*sqrt(1/2*sqrt(5) + 5/2)
+            1/4*sqrt(2*sqrt(5) + 10)
             sage: cos(pi/12)
-            1/12*sqrt(6)*(sqrt(3) + 3)
+            1/4*sqrt(6) + 1/4*sqrt(2)
             sage: cos(pi/15)
             1/8*sqrt(5) + 1/4*sqrt(3/2*sqrt(5) + 15/2) - 1/8
             sage: cos(pi/24)
             1/4*sqrt(2*sqrt(6) + 2*sqrt(2) + 8)
+            sage: cos(104*pi/105)
+            -cos(1/105*pi)
             sage: tan(pi/5)
             sqrt(-2*sqrt(5) + 5)
             sage: tan(pi/8)
             sqrt(2) - 1
             sage: tan(pi/10)
-            sqrt(-2/5*sqrt(5) + 1)
+            1/5*sqrt(-10*sqrt(5) + 25)
             sage: tan(pi/16)
             -sqrt(2) + sqrt(2*sqrt(2) + 4) - 1
             sage: tan(pi/20)
-            sqrt(5) - 1/2*sqrt(8*sqrt(5) + 20) + 1
+            sqrt(5) - sqrt(2*sqrt(5) + 5) + 1
             sage: tan(pi/24)
             sqrt(6) - sqrt(3) + sqrt(2) - 2
+            sage: tan(104*pi/105)
+            -tan(1/105*pi)
+            sage: cot(104*pi/105)
+            -cot(1/105*pi)
+            sage: sec(104*pi/105)
+            -sec(1/105*pi)
+            sage: csc(104*pi/105)
+            csc(1/105*pi)
 
             sage: all(sin(rat*pi).n(200)-sin(rat*pi,hold=True).n(200) < 1e-30 for rat in [1/5,2/5,1/30,7/30,11/30,13/30,1/8,3/8,1/24,5/24,7/24,11/24])
             True
             sage: all(cos(rat*pi).n(200)-cos(rat*pi,hold=True).n(200) < 1e-30 for rat in [1/10,3/10,1/12,5/12,1/15,2/15,4/15,7/15,1/8,3/8,1/24,5/24,11/24])
             True
             sage: all(tan(rat*pi).n(200)-tan(rat*pi,hold=True).n(200) < 1e-30 for rat in [1/5,2/5,1/10,3/10,1/20,3/20,7/20,9/20,1/8,3/8,1/16,3/16,5/16,7/16,1/24,5/24,7/24,11/24])
             True
+
+        Check that :trac:`20456` is fixed::
+
+            sage: assume(x>0)
+            sage: sin(pi*x)
+            sin(pi*x)
+            sage: forget()
         """
         GinacFunction.__init__(self, "sin", latex_name=r"\sin",
                 conversions=dict(maxima='sin',mathematica='Sin'))

src/sage/modular/modform_hecketriangle/hecke_triangle_groups.py

@@ -541,7 +541,7 @@ def dvalue(self):
             sage: HeckeTriangleGroup(6).dvalue()
             1/108
             sage: HeckeTriangleGroup(10).dvalue()
-            e^(2*euler_gamma - 2*pi/sqrt(1/2*sqrt(5) + 5/2) + psi(4/5) + psi(7/10))
+            e^(2*euler_gamma - 4*pi/sqrt(2*sqrt(5) + 10) + psi(4/5) + psi(7/10))
             sage: HeckeTriangleGroup(infinity).dvalue()
             1/64
         """

src/sage/modular/modform_hecketriangle/readme.py

@@ -37,7 +37,7 @@
       sage: G.is_arithmetic()
       False
       sage: G.dvalue()
-      e^(2*euler_gamma - 2*sqrt(6)*pi/(sqrt(3) + 3) + psi(19/24) + psi(17/24))
+      e^(2*euler_gamma - 4*pi/(sqrt(6) + sqrt(2)) + psi(19/24) + psi(17/24))
       sage: AA(G.lam())
       1.9318516525781...?
 

src/sage/symbolic/expression.pyx

@@ -3584,6 +3584,22 @@ cdef class Expression(CommutativeRingElement):
             sage: latex(ex)
             \left(e^{\sqrt{x}}\right)^{x}
 
+        Test simplification of powers involving the reciprocal
+        logarithm of the (positive) base::
+
+            sage: 2^(1/log(2))
+            e
+            sage: 2^(x/log(2))
+            e^x
+            sage: 2^(-x^2/2/log(2))
+            e^(-1/2*x^2)
+            sage: x^(x/log(x))
+            x^(x/log(x))
+            sage: assume(x > 0)
+            sage: x^(x/log(x))
+            e^x
+            sage: forget()
+
         Test base a Python numeric type::
 
             sage: int(2)^x

src/sage/symbolic/pynac.pyx

@@ -1100,6 +1100,14 @@ cdef bint py_is_real(object a) except +:
     if type(a) is int or isinstance(a, Integer) or\
             type(a) is long or type(a) is float:
         return True
+    try:
+        P = parent_c(a)
+        if P.is_field() and P.is_finite():
+            return False
+    except NotImplementedError:
+        return False
+    except AttributeError:
+        pass
     return py_imag(a) == 0
 
 cdef bint py_is_prime(object n) except +:

src/sage/symbolic/ring.pyx

@@ -293,6 +293,14 @@ cdef class SymbolicRing(CommutativeRing):
             3*x^5*log(y)
             sage: t.operator(), t.operands()
             (<function mul_vararg at 0x...>, [x^5, log(y), 3])
+
+        Check that :trac:`20162` is fixed::
+
+            sage: k.<a> = GF(9)
+            sage: SR(a).is_real()
+            False
+            sage: SR(a).is_positive()
+            False
         """
         cdef GEx exp
         if is_Expression(x):