Ticket #7357: li(x).patch

sage/functions/all.py

@@ -26 +26 @@
 from transcendental import (exponential_integral_1,
                             gamma_inc, incomplete_gamma,
                             zeta, zeta_symmetric,
-                            Li, Ei, 
+                            Li, li, Ei, 
                             dickman_rho)
 
 from special import (bessel_I, bessel_J, bessel_K, bessel_Y,

sage/functions/transcendental.py

@@ -24 +24 @@
 import sage.rings.complex_field as complex_field
 import sage.rings.real_double as real_double
 import sage.rings.complex_number
-from sage.gsl.integration import numerical_integral
 
 from sage.rings.all import (is_RealNumber, RealField,
                             is_ComplexNumber, ComplexField,
@@ -344 +343 @@
     finally:
         mp.mp.prec = old_prec
     return mp.mpmath_to_sage(mp_ret,prec)
+    
+def li(x, prec = None):
+    r"""
+    Computes li(x).
+    
+    This is the function
+    
+    .. math::
+    
+                mathrm{li}(x) = int_0^x frac{1}{log t} , dt     
+                
+    Note that li has a singularity at x = 1 and will return -infinity and 
+    that this function is different from Li(x), the offset logarithmic
+    integral.
+    
+    
+    ALGORITHM: Computed using mpmath.
+    
+    INPUT:
+    
+    
+    -  ``x`` - a number.
+    
+    
+    OUTPUT:
+    
+    
+    -  ``li(x)`` - a number
+    
+    
+    EXAMPLES::
+    
+        sage: li(2)
+        1.04516378011749
+        sage: li(2)
+        1.04516378011749
+        sage: li(5)
+        3.63458831003265
+        sage: li(1000)
+        177.609657990152
+        sage: li(10^5)
+        9629.80900105080
+        sage: prime_pi(10^5)
+        9592
+        sage: li(1)
+        -infinity
+    
+    """
+    if prec == None:
+        try:
+            prec = x.prec()
+        except AttributeError:
+            prec = 53
+    import sage.libs.mpmath.all as mp
+    return mp.call(mp.li,x,prec=prec)
 
 import math
 def _one_over_log(t):