diff --git a/include/boost/math/special_functions.hpp b/include/boost/math/special_functions.hpp
index 3468af2b6..f33f985e2 100644
--- a/include/boost/math/special_functions.hpp
+++ b/include/boost/math/special_functions.hpp
@@ -27,6 +27,7 @@
 #include <boost/math/special_functions/ellint_1.hpp>
 #include <boost/math/special_functions/ellint_2.hpp>
 #include <boost/math/special_functions/ellint_3.hpp>
+#include <boost/math/special_functions/ellint_d.hpp>
 #include <boost/math/special_functions/ellint_rc.hpp>
 #include <boost/math/special_functions/ellint_rd.hpp>
 #include <boost/math/special_functions/ellint_rf.hpp>
diff --git a/include/boost/math/special_functions/ellint_d.hpp b/include/boost/math/special_functions/ellint_d.hpp
index 0e1074a09..5bd065d6e 100644
--- a/include/boost/math/special_functions/ellint_d.hpp
+++ b/include/boost/math/special_functions/ellint_d.hpp
@@ -11,8 +11,8 @@
 //  that the code continues to work no matter how many digits
 //  type T has.
 
-#ifndef BOOST_MATH_ELLINT_2_HPP
-#define BOOST_MATH_ELLINT_2_HPP
+#ifndef BOOST_MATH_ELLINT_D_HPP
+#define BOOST_MATH_ELLINT_D_HPP
 
 #ifdef _MSC_VER
 #pragma once
@@ -122,6 +122,8 @@ T ellint_d_imp(T k, const Policy& pol)
     {
         return static_cast<T>(1);
     }
+    if(fabs(k) <= tools::root_epsilon<T>())
+       return constants::pi<T>() / 4;
 
     T x = 0;
     T t = k * k;
@@ -174,5 +176,5 @@ inline typename tools::promote_args<T1, T2>::type ellint_d(T1 k, T2 phi, const P
 
 }} // namespaces
 
-#endif // BOOST_MATH_ELLINT_2_HPP
+#endif // BOOST_MATH_ELLINT_D_HPP
 
diff --git a/include/boost/math/special_functions/math_fwd.hpp b/include/boost/math/special_functions/math_fwd.hpp
index a350e44dc..d9586fa2d 100644
--- a/include/boost/math/special_functions/math_fwd.hpp
+++ b/include/boost/math/special_functions/math_fwd.hpp
@@ -330,6 +330,15 @@ namespace boost
    template <class T1, class T2, class Policy>
    typename tools::promote_args<T1, T2>::type ellint_1(T1 k, T2 phi, const Policy& pol);
 
+   template <typename T>
+   typename tools::promote_args<T>::type ellint_d(T k);
+
+   template <class T1, class T2>
+   typename tools::promote_args<T1, T2>::type ellint_d(T1 k, T2 phi);
+
+   template <class T1, class T2, class Policy>
+   typename tools::promote_args<T1, T2>::type ellint_d(T1 k, T2 phi, const Policy& pol);
+
    namespace detail{
 
    template <class T, class U, class V>
@@ -1167,6 +1176,12 @@ namespace boost
 \
    template <class T1, class T2>\
    inline typename boost::math::tools::promote_args<T1, T2>::type ellint_2(T1 k, T2 phi){ return boost::math::ellint_2(k, phi, Policy()); }\
+\
+   template <typename T>\
+   inline typename boost::math::tools::promote_args<T>::type ellint_d(T k){ return boost::math::ellint_d(k, Policy()); }\
+\
+   template <class T1, class T2>\
+   inline typename boost::math::tools::promote_args<T1, T2>::type ellint_d(T1 k, T2 phi){ return boost::math::ellint_d(k, phi, Policy()); }\
 \
    template <typename T>\
    inline typename boost::math::tools::promote_args<T>::type ellint_1(T k){ return boost::math::ellint_1(k, Policy()); }\
diff --git a/test/compile_test/instantiate.hpp b/test/compile_test/instantiate.hpp
index 2b77707d0..c008b556c 100644
--- a/test/compile_test/instantiate.hpp
+++ b/test/compile_test/instantiate.hpp
@@ -239,6 +239,7 @@ void instantiate(RealType)
    boost::math::ellint_2(v1, v2);
    boost::math::ellint_3(v1, v2);
    boost::math::ellint_3(v1, v2, v3);
+   boost::math::ellint_d(v1);
    boost::math::ellint_rc(v1, v2);
    boost::math::ellint_rd(v1, v2, v3);
    boost::math::ellint_rf(v1, v2, v3);
@@ -432,6 +433,8 @@ void instantiate(RealType)
    boost::math::ellint_rf(v1 * 1, v2 + 0, v3 / 1);
    boost::math::ellint_rg(v1 * 1, v2 + 0, v3 / 1);
    boost::math::ellint_rj(v1 * 1, v2 + 0, v3 / 1, v1 * 1);
+   boost::math::ellint_d(v1 * 1);
+   boost::math::ellint_d(v1 * 1, v2 + 0);
    boost::math::jacobi_elliptic(v1 * 1, v2 + 0, &v1, &v2);
    boost::math::jacobi_cd(v1 * 1, v2 + 0);
    boost::math::jacobi_cn(v1 * 1, v2 + 0);
@@ -595,6 +598,8 @@ void instantiate(RealType)
    boost::math::ellint_2(v1, v2, pol);
    boost::math::ellint_3(v1, v2, pol);
    boost::math::ellint_3(v1, v2, v3, pol);
+   boost::math::ellint_d(v1, pol);
+   boost::math::ellint_d(v1, v2, pol);
    boost::math::ellint_rc(v1, v2, pol);
    boost::math::ellint_rd(v1, v2, v3, pol);
    boost::math::ellint_rf(v1, v2, v3, pol);
@@ -785,6 +790,8 @@ void instantiate(RealType)
    test::ellint_2(v1, v2);
    test::ellint_3(v1, v2);
    test::ellint_3(v1, v2, v3);
+   test::ellint_d(v1);
+   test::ellint_d(v1, v2);
    test::ellint_rc(v1, v2);
    test::ellint_rd(v1, v2, v3);
    test::ellint_rf(v1, v2, v3);
@@ -991,6 +998,9 @@ void instantiate_mixed(RealType)
    boost::math::ellint_3(fr, lr);
    boost::math::ellint_3(s, l, i);
    boost::math::ellint_3(fr, dr, lr);
+   boost::math::ellint_d(i);
+   boost::math::ellint_d(i, l);
+   boost::math::ellint_d(fr, lr);
    boost::math::ellint_rc(i, s);
    boost::math::ellint_rc(fr, lr);
    boost::math::ellint_rd(s, i, l);
@@ -1132,6 +1142,9 @@ void instantiate_mixed(RealType)
    boost::math::ellint_3(fr, lr, pol);
    boost::math::ellint_3(s, l, i, pol);
    boost::math::ellint_3(fr, dr, lr, pol);
+   boost::math::ellint_d(i, pol);
+   boost::math::ellint_d(i, l, pol);
+   boost::math::ellint_d(fr, lr, pol);
    boost::math::ellint_rc(i, s, pol);
    boost::math::ellint_rc(fr, lr, pol);
    boost::math::ellint_rd(s, i, l, pol);
@@ -1276,6 +1289,9 @@ void instantiate_mixed(RealType)
    test::ellint_3(fr, lr);
    test::ellint_3(s, l, i);
    test::ellint_3(fr, dr, lr);
+   test::ellint_d(i);
+   test::ellint_d(i, l);
+   test::ellint_d(fr, lr);
    test::ellint_rc(i, s);
    test::ellint_rc(fr, lr);
    test::ellint_rd(s, i, l);
diff --git a/test/compile_test/sf_ellint_d_incl_test.cpp b/test/compile_test/sf_ellint_d_incl_test.cpp
new file mode 100644
index 000000000..c7c4249a5
--- /dev/null
+++ b/test/compile_test/sf_ellint_d_incl_test.cpp
@@ -0,0 +1,29 @@
+//  Copyright John Maddock 2006.
+//  Use, modification and distribution are subject to the
+//  Boost Software License, Version 1.0. (See accompanying file
+//  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+//
+// Basic sanity check that header <boost/math/special_functions/ellint_d.hpp>
+// #includes all the files that it needs to.
+//
+#include <boost/math/special_functions/ellint_d.hpp>
+//
+// Note this header includes no other headers, this is
+// important if this test is to be meaningful:
+//
+#include "test_compile_result.hpp"
+
+void compile_and_link_test()
+{
+   check_result<float>(boost::math::ellint_d<float>(f, f));
+   check_result<double>(boost::math::ellint_d<double>(d, d));
+#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
+   check_result<long double>(boost::math::ellint_d<long double>(l, l));
+#endif
+
+   check_result<float>(boost::math::ellint_d<float>(f));
+   check_result<double>(boost::math::ellint_d<double>(d));
+#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
+   check_result<long double>(boost::math::ellint_d<long double>(l));
+#endif
+}
diff --git a/test/test_ellint_d.hpp b/test/test_ellint_d.hpp
index a31b11ee0..88e7ac193 100644
--- a/test/test_ellint_d.hpp
+++ b/test/test_ellint_d.hpp
@@ -13,7 +13,6 @@
 #include <boost/test/unit_test.hpp>
 #include <boost/test/floating_point_comparison.hpp>
 #include <boost/math/special_functions/math_fwd.hpp>
-#include <boost/math/special_functions/ellint_d.hpp>
 #include <boost/array.hpp>
 #include "functor.hpp"
 
@@ -96,9 +95,10 @@ void test_spots(T, const char* type_name)
 
     // Function values calculated on http://functions.wolfram.com/
     // Note that Mathematica's EllipticE accepts k^2 as the second parameter.
-    static const boost::array<boost::array<T, 2>, 2> data2 = {{
+    static const boost::array<boost::array<T, 2>, 3> data2 = {{
        { { SC_(0.5), SC_(0.87315258189267554964563356323264341) } },
        { { SC_(1.0) / 1024, SC_(0.78539844427788694671464428063604776) } },
+       { { boost::math::tools::root_epsilon<T>(), SC_(0.78539816339744830961566084581987572) } }
     }};
 
     do_test_ellint_d1<T>(data2, type_name, "Elliptic Integral E: Mathworld Data");
diff --git a/test/test_instances/double_test_instances_4.cpp b/test/test_instances/double_test_instances_4.cpp
index 437bb5bdb..f78783d92 100644
--- a/test/test_instances/double_test_instances_4.cpp
+++ b/test/test_instances/double_test_instances_4.cpp
@@ -11,6 +11,7 @@
 #include <boost/math/special_functions/ellint_1.hpp>
 #include <boost/math/special_functions/ellint_2.hpp>
 #include <boost/math/special_functions/ellint_3.hpp>
+#include <boost/math/special_functions/ellint_d.hpp>
 #include <boost/math/special_functions/ellint_rc.hpp>
 #include <boost/math/special_functions/ellint_rf.hpp>
 #include <boost/math/special_functions/ellint_rj.hpp>
diff --git a/test/test_instances/float_test_instances_4.cpp b/test/test_instances/float_test_instances_4.cpp
index 0be679981..3dd89365d 100644
--- a/test/test_instances/float_test_instances_4.cpp
+++ b/test/test_instances/float_test_instances_4.cpp
@@ -11,6 +11,7 @@
 #include <boost/math/special_functions/ellint_1.hpp>
 #include <boost/math/special_functions/ellint_2.hpp>
 #include <boost/math/special_functions/ellint_3.hpp>
+#include <boost/math/special_functions/ellint_d.hpp>
 #include <boost/math/special_functions/ellint_rc.hpp>
 #include <boost/math/special_functions/ellint_rf.hpp>
 #include <boost/math/special_functions/ellint_rj.hpp>
diff --git a/test/test_instances/ldouble_test_instances_4.cpp b/test/test_instances/ldouble_test_instances_4.cpp
index 5d2fc9ad2..e95c06bd8 100644
--- a/test/test_instances/ldouble_test_instances_4.cpp
+++ b/test/test_instances/ldouble_test_instances_4.cpp
@@ -11,6 +11,7 @@
 #include <boost/math/special_functions/ellint_1.hpp>
 #include <boost/math/special_functions/ellint_2.hpp>
 #include <boost/math/special_functions/ellint_3.hpp>
+#include <boost/math/special_functions/ellint_d.hpp>
 #include <boost/math/special_functions/ellint_rc.hpp>
 #include <boost/math/special_functions/ellint_rf.hpp>
 #include <boost/math/special_functions/ellint_rj.hpp>
diff --git a/test/test_instances/real_concept_test_instances_4.cpp b/test/test_instances/real_concept_test_instances_4.cpp
index 36dbd7e0d..b6b66b025 100644
--- a/test/test_instances/real_concept_test_instances_4.cpp
+++ b/test/test_instances/real_concept_test_instances_4.cpp
@@ -11,6 +11,7 @@
 #include <boost/math/special_functions/ellint_1.hpp>
 #include <boost/math/special_functions/ellint_2.hpp>
 #include <boost/math/special_functions/ellint_3.hpp>
+#include <boost/math/special_functions/ellint_d.hpp>
 #include <boost/math/special_functions/ellint_rc.hpp>
 #include <boost/math/special_functions/ellint_rf.hpp>
 #include <boost/math/special_functions/ellint_rj.hpp>
diff --git a/test/test_instances/test_instances.hpp b/test/test_instances/test_instances.hpp
index 58dec8d91..138231193 100644
--- a/test/test_instances/test_instances.hpp
+++ b/test/test_instances/test_instances.hpp
@@ -218,6 +218,12 @@ namespace boost{ namespace math{
 
    template tools::promote_args<BOOST_MATH_TEST_TYPE, BOOST_MATH_TEST_TYPE>::type ellint_2(BOOST_MATH_TEST_TYPE k, BOOST_MATH_TEST_TYPE phi, const policies::policy<>& pol);
 
+   template tools::promote_args<BOOST_MATH_TEST_TYPE>::type ellint_d(BOOST_MATH_TEST_TYPE k);
+
+   template tools::promote_args<BOOST_MATH_TEST_TYPE, BOOST_MATH_TEST_TYPE>::type ellint_d(BOOST_MATH_TEST_TYPE k, BOOST_MATH_TEST_TYPE phi);
+
+   template tools::promote_args<BOOST_MATH_TEST_TYPE, BOOST_MATH_TEST_TYPE>::type ellint_d(BOOST_MATH_TEST_TYPE k, BOOST_MATH_TEST_TYPE phi, const policies::policy<>& pol);
+
    template tools::promote_args<BOOST_MATH_TEST_TYPE>::type ellint_1(BOOST_MATH_TEST_TYPE k);
 
    template tools::promote_args<BOOST_MATH_TEST_TYPE, BOOST_MATH_TEST_TYPE>::type ellint_1(BOOST_MATH_TEST_TYPE k, BOOST_MATH_TEST_TYPE phi);