Merge branch 'u/rws/17678-1' of git://trac.sagemath.org/sage into tes…

…t_avoid_recompilation

Conflicts:
	src/sage/symbolic/ring.pyx
	src/sage/tests/french_book/recequadiff.py

src/sage/functions/bessel.py

@@ -125,7 +125,7 @@
         sage: bessel_J(0, x)
         bessel_J(0, x)
         sage: bessel_J(0, 0)
-        bessel_J(0, 0)
+        1
         sage: bessel_J(0, x).diff(x)
         -1/2*bessel_J(1, x) + 1/2*bessel_J(-1, x)
 
@@ -224,11 +224,12 @@
 
 # remove after deprecation period
 from sage.calculus.calculus import maxima
-from sage.functions.other import real, imag
+from sage.functions.trig import sin, cos
+from sage.functions.other import real, imag, sqrt
 from sage.misc.sage_eval import sage_eval
 from sage.rings.real_mpfr import RealField
 from sage.plot.plot import plot
-from sage.rings.all import ZZ
+from sage.rings.all import ZZ, QQ
 
 
 class Function_Bessel_J(BuiltinFunction):
@@ -341,6 +342,37 @@ def __init__(self):
                                                   maxima='bessel_j',
                                                   sympy='besselj'))
 
+    def _eval_(self, n, x):
+        """
+        EXAMPLES::
+
+            sage: n = var('n')
+            sage: bessel_J(0, 0)
+            1
+            sage: bessel_J(5/2, 0)
+            0
+            sage: bessel_J(-5/2, 0)
+            Infinity
+            sage: bessel_J(1/2, x)
+            sqrt(2)*sqrt(1/(pi*x))*sin(x)
+            sage: bessel_J(-1/2, x)
+            sqrt(2)*sqrt(1/(pi*x))*cos(x)
+            sage: bessel_J(n, 0)
+            bessel_J(n, 0)
+        """
+        from sage.rings.infinity import unsigned_infinity
+        if x == 0:
+            if n == 0:
+                return ZZ(1)
+            elif n > 0 or n in ZZ:
+                return ZZ(0)
+            elif not isinstance(n, Expression):
+                return unsigned_infinity
+        elif n == QQ(1)/2:
+            return sqrt(2/pi/x) * sin(x)
+        elif n == QQ(-1)/2:
+            return sqrt(2/pi/x) * cos(x)
+
     def _evalf_(self, n, x, parent=None, algorithm=None):
         """
         EXAMPLES::
@@ -521,6 +553,29 @@ def __init__(self):
                                                   maxima='bessel_y',
                                                   sympy='bessely'))
 
+    def _eval_(self, n, x):
+        """
+        EXAMPLES::
+
+            sage: n = var('n')
+            sage: bessel_Y(1, 0)
+            Infinity
+            sage: bessel_Y(1/2, x)
+            -sqrt(2)*sqrt(1/(pi*x))*cos(x)
+            sage: bessel_Y(-1/2, x)
+            sqrt(2)*sqrt(1/(pi*x))*sin(x)
+        """
+        from sage.rings.infinity import infinity, unsigned_infinity
+        if x == 0:
+            if n == 0:
+                return -infinity
+            elif not isinstance(n, Expression):
+                return unsigned_infinity
+        elif n == QQ(1)/2:
+            return -sqrt(2/pi/x) * cos(x)
+        elif n == QQ(-1)/2:
+            return sqrt(2/pi/x) * sin(x)
+
     def _evalf_(self, n, x, parent=None, algorithm=None):
         """
         EXAMPLES::
@@ -698,26 +753,43 @@ def _eval_(self, n, x):
         """
         EXAMPLES::
 
-            sage: y=var('y')
-            sage: bessel_I(y,x)
+            sage: n,y = var('n,y')
+            sage: bessel_I(y, x)
             bessel_I(y, x)
-            sage: bessel_I(0.0, 1.0)
-            1.26606587775201
+            sage: bessel_I(0, 0)
+            1
+            sage: bessel_I(7/2, 0)
+            0
+            sage: bessel_I(-7/2, 0)
+            Infinity
             sage: bessel_I(1/2, 1)
             sqrt(2)*sinh(1)/sqrt(pi)
             sage: bessel_I(-1/2, pi)
             sqrt(2)*cosh(pi)/pi
+            sage: bessel_I(n, 0)
+            bessel_I(n, 0)
         """
+        from sage.rings.infinity import unsigned_infinity
         # special identities
-        if n == Integer(1) / Integer(2):
+        if x == 0:
+            if n == 0:
+                return ZZ(1)
+            elif n > 0 or n in ZZ:
+                return ZZ(0)
+            elif not isinstance(n, Expression):
+                return unsigned_infinity
+        if n == QQ(1)/2:
             return sqrt(2 / (pi * x)) * sinh(x)
-        elif n == -Integer(1) / Integer(2):
+        elif n == -QQ(1)/2:
             return sqrt(2 / (pi * x)) * cosh(x)
 
+
     def _evalf_(self, n, x, parent=None, algorithm=None):
         """
         EXAMPLES::
 
+            sage: bessel_I(0.0, 1.0)
+            1.26606587775201
             sage: bessel_I(1,3).n(digits=20)
             3.9533702174026093965
         """
@@ -793,9 +865,9 @@ class Function_Bessel_K(BuiltinFunction):
         -1/2*bessel_K(3, x) - 1/2*bessel_K(1, x)
 
         sage: bessel_K(1/2, x)
-        bessel_K(1/2, x)
+        sqrt(1/2)*sqrt(pi)*e^(-x)/sqrt(x)
         sage: bessel_K(1/2, -1)
-        bessel_K(1/2, -1)
+        -I*sqrt(1/2)*sqrt(pi)*e
         sage: bessel_K(1/2, 1)
         sqrt(1/2)*sqrt(pi)*e^(-1)
 
@@ -879,23 +951,30 @@ def _eval_(self, n, x):
         """
         EXAMPLES::
 
-            sage: bessel_K(1,0)
-            bessel_K(1, 0)
-            sage: bessel_K(1.0, 0.0)
-            +infinity
-            sage: bessel_K(-1, 1).n(128)
-            0.60190723019723457473754000153561733926
-        """
+            sage: n = var('n')
+            sage: bessel_K(1, 0)
+            Infinity
+            sage: bessel_K(1/2, x)
+            sqrt(1/2)*sqrt(pi)*e^(-x)/sqrt(x)
+            sage: bessel_K(n, 0)
+            bessel_K(n, 0)
+        """
+        from sage.rings.infinity import unsigned_infinity
         # special identity
-        if n == Integer(1) / Integer(2) and x > 0:
+        if x == 0 and not isinstance(n, Expression):
+            return unsigned_infinity
+        if n == QQ(1)/2 or n == -QQ(1)/2 and x > 0:
             return sqrt(pi / 2) * exp(-x) * x ** (-Integer(1) / Integer(2))
 
+
     def _evalf_(self, n, x, parent=None, algorithm=None):
         """
         EXAMPLES::
 
             sage: bessel_K(0.0, 1.0)
             0.421024438240708
+            sage: bessel_K(-1, 1).n(128)
+            0.60190723019723457473754000153561733926
             sage: bessel_K(0, RealField(128)(1))
             0.42102443824070833333562737921260903614
         """

src/sage/rings/infinity.py

@@ -481,7 +481,7 @@ def _div_(self, other):
             sage: SR(infinity) / unsigned_infinity
             Traceback (most recent call last):
             ...
-            ValueError: unsigned oo times smaller number not defined
+            RuntimeError: indeterminate expression: 0 * infinity encountered.
         """
         return self * ~other
 

src/sage/symbolic/expression.pyx

@@ -3067,7 +3067,7 @@ cdef class Expression(CommutativeRingElement):
             sage: x*unsigned_infinity
             Traceback (most recent call last):
             ...
-            ValueError: oo times number < oo not defined
+            RuntimeError: indeterminate expression: infinity * f(x) encountered.
 
             sage: SR(oo)*SR(oo)
             +Infinity

src/sage/symbolic/ring.pyx

@@ -96,6 +96,8 @@ cdef class SymbolicRing(CommutativeRing):
             2
             sage: SR.coerce(-infinity)
             -Infinity
+            sage: SR.coerce(unsigned_infinity)
+            Infinity
             sage: SR.has_coerce_map_from(ZZ['t'])
             True
             sage: SR.has_coerce_map_from(ZZ['t,u,v'])
@@ -114,6 +116,8 @@ cdef class SymbolicRing(CommutativeRing):
             True
             sage: SR.has_coerce_map_from(ComplexBallField())
             True
+            sage: SR.has_coerce_map_from(UnsignedInfinityRing)
+            True
 
         TESTS:
 
@@ -177,7 +181,8 @@ cdef class SymbolicRing(CommutativeRing):
             from sage.rings.polynomial.laurent_polynomial_ring import is_LaurentPolynomialRing
 
             from sage.rings.all import (ComplexField,
-                                        RLF, CLF, AA, QQbar, InfinityRing)
+                                        RLF, CLF, AA, QQbar, InfinityRing,
+                                        UnsignedInfinityRing)
             from sage.rings.finite_rings.finite_field_base import is_FiniteField
 
             from sage.interfaces.maxima import Maxima
@@ -190,7 +195,7 @@ cdef class SymbolicRing(CommutativeRing):
             elif is_PolynomialRing(R) or is_MPolynomialRing(R) or is_FractionField(R) or is_LaurentPolynomialRing(R):
                 base = R.base_ring()
                 return base is not self and self.has_coerce_map_from(base)
-            elif (R is InfinityRing
+            elif (R is InfinityRing or R is UnsignedInfinityRing
                   or is_RealIntervalField(R) or is_ComplexIntervalField(R)
                   or isinstance(R, RealBallField)
                   or isinstance(R, ComplexBallField)

src/sage/tests/french_book/recequadiff.py

@@ -12,6 +12,9 @@
 
   sage: x = var('x')
   sage: y = function('y')(x)
+  sage: _C = SR.var("_C")
+  sage: _K1 = SR.var("_K1")
+  sage: _K2 = SR.var("_K2")
 
 Sage example in ./recequadiff.tex, line 179::
 
@@ -104,10 +107,8 @@
 
 Sage example in ./recequadiff.tex, line 367::
 
-  sage: solve(ed, y)[0].subs(_C==5).rhs()
-  Traceback (most recent call last):
-  ...
-  NameError: name '_C' is not defined
+  sage: solve(ed, y)[0].subs(_C == 5).rhs()
+  e^(-sqrt(-2*cos(x) + 10))
 
 Sage example in ./recequadiff.tex, line 377::
 
@@ -129,16 +130,16 @@
 
   sage: P = Graphics()
   sage: for k in range(1,20,2):
-  ...       P += plot(solve(ed, y)[0].subs(c==1+k/4).rhs(), x, -3, 3)
+  ...       P += plot(solve(ed, y)[0].subs(c == 1+k/4).rhs(), x, -3, 3)
   sage: P
-  Graphics object consisting of 11 graphics primitives
+  Graphics object consisting of 10 graphics primitives
 
 Sage example in ./recequadiff.tex, line 426::
 
   sage: P = Graphics()
   sage: for j in [0,1]:
   ...       for k in range(1,10,2):
-  ...           f = solve(ed,y)[j].subs(c==2+0.25*k).rhs()
+  ...           f = solve(ed,y)[j].subs(c == 2+0.25*k).rhs()
   ...           P += plot(f, x, -3, 3)
   sage: P
   Graphics object consisting of 10 graphics primitives