a few minor details in special_values.py

src/sage/quadratic_forms/special_values.py

@@ -13,7 +13,6 @@
                              factorial,
                              fundamental_discriminant,
                              kronecker as kronecker_symbol)
-from sage.misc.functional import denominator
 from sage.rings.infinity import infinity
 from sage.rings.integer_ring import ZZ
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
@@ -24,7 +23,7 @@
 
 def gamma__exact(n):
     r"""
-    Evaluates the exact value of the `\Gamma` function at an integer or
+    Return the exact value of the `\Gamma` function at an integer or
     half-integer argument.
 
     EXAMPLES::
@@ -67,14 +66,13 @@ def gamma__exact(n):
     """
     n = QQ(n)
 
-    if denominator(n) == 1:
+    if n.denominator() == 1:
         if n <= 0:
             return infinity
         return factorial(n - 1)
 
-    if denominator(n) == 2:
+    if n.denominator() == 2:
         # now n = 1/2 + an integer
-        from sage.misc.functional import sqrt
         from sage.symbolic.constants import pi
 
         ans = QQ.one()
@@ -86,8 +84,7 @@ def gamma__exact(n):
                 n += -1
                 ans *= n
 
-        ans *= sqrt(pi)
-        return ans
+        return ans * pi.sqrt()
 
     raise TypeError("you must give an integer or half-integer argument")
 
@@ -98,8 +95,8 @@ def zeta__exact(n):
     r"""
     Return the exact value of the Riemann Zeta function.
 
-    The argument must be a critical value, namely either positive even
-    or negative odd.
+    The argument `n` must be an integer which is a critical value,
+    namely either positive even or negative odd.
 
     See for example [Iwa1972]_, p13, Special value of `\zeta(2k)`
 
@@ -147,18 +144,14 @@ def zeta__exact(n):
     - [Was1997]_
     """
     if n < 0:
-        return bernoulli(1-n)/(n-1)
-    elif n > 1:
-        if (n % 2 == 0):
+        return bernoulli(1 - n) / (n - 1)
+    if n > 1:
+        if not n % 2:
             from sage.symbolic.constants import pi
 
-            return ZZ(-1)**(n//2 + 1) * ZZ(2)**(n-1) * pi**n * bernoulli(n) / factorial(n)
-        else:
-            raise TypeError("n must be a critical value (i.e. even > 0 or odd < 0)")
-    elif n == 1:
-        return infinity
-    elif n == 0:
-        return QQ((-1, 2))
+            return ZZ(-1)**(n // 2 + 1) * ZZ(2)**(n - 1) * pi**n * bernoulli(n) / factorial(n)
+        raise TypeError("n must be a critical value (i.e. even > 0 or odd < 0)")
+    return infinity if n == 1 else QQ((-1, 2))
 
 
 # ---------- Dirichlet L-functions with quadratic characters ----------
@@ -209,7 +202,7 @@ def quadratic_L_function__exact(n, d):
     Return the exact value of a quadratic twist of the Riemann Zeta function
     by `\chi_d(x) = \left(\frac{d}{x}\right)`.
 
-    The input `n` must be a critical value.
+    The input `n` must be an integer which is a critical value.
 
     EXAMPLES::
 
@@ -234,36 +227,30 @@ def quadratic_L_function__exact(n, d):
     - [Was1997]_
     """
     if n <= 0:
-        return QuadraticBernoulliNumber(1-n, d)/(n-1)
-    elif n >= 1:
-        # Compute the kind of critical values (p10)
-        if kronecker_symbol(fundamental_discriminant(d), -1) == 1:
-            delta = 0
-        else:
-            delta = 1
-
-        # Compute the positive special values (p17)
-        if ((n - delta) % 2 == 0):
-            from sage.misc.functional import sqrt
-            from sage.symbolic.constants import I, pi
-            from sage.symbolic.ring import SR
-
-            f = abs(fundamental_discriminant(d))
-            if delta == 0:
-                GS = sqrt(f)
-            else:
-                GS = I * sqrt(f)
-            ans = SR(ZZ(-1)**(1+(n-delta)/2))
-            ans *= (2*pi/f)**n
-            ans *= GS     # Evaluate the Gauss sum here! =0
-            ans *= QQ.one()/(2 * I**delta)
-            ans *= QuadraticBernoulliNumber(n, d)/factorial(n)
-            return ans
-        else:
-            if delta == 0:
-                raise TypeError("n must be a critical value (i.e. even > 0 or odd < 0)")
-            if delta == 1:
-                raise TypeError("n must be a critical value (i.e. odd > 0 or even <= 0)")
+        return QuadraticBernoulliNumber(1 - n, d) / (n - 1)
+
+    # Compute the kind of critical values (p10)
+    if kronecker_symbol(fundamental_discriminant(d), -1) == 1:
+        delta = 0
+    else:
+        delta = 1
+
+    # Compute the positive special values (p17)
+    if not (n - delta) % 2:
+        from sage.symbolic.constants import I, pi
+
+        f = abs(fundamental_discriminant(d))
+        GS = f.sqrt() if delta == 0 else I * f.sqrt()
+        ans = (2 * pi / f)**n
+        ans *= ZZ(-1)**(1 + (n - delta) // 2)
+        ans *= GS     # Evaluate the Gauss sum here! =0
+        ans *= QQ.one() / (2 * I**delta)
+        ans *= QuadraticBernoulliNumber(n, d) / factorial(n)
+        return ans
+
+    if delta == 0:
+        raise TypeError("n must be a critical value (i.e. even > 0 or odd < 0)")
+    raise TypeError("n must be a critical value (i.e. odd > 0 or even <= 0)")
 
 
 def quadratic_L_function__numerical(n, d, num_terms=1000):