Checked Bessel function outcome against online calculator, and removed 'TODO' tag.

Tests/UnitTests/Core/SpecialFunctionsTest.h

@@ -34,26 +34,29 @@ TEST_F(SpecialFunctionsTest, BesselJ1)
     const double eps = 4.7e-16; // more than twice the machine precision
     complex_t res;
 
-    // Test against values computed with dev-tools/math/cbesselJ01.c
-    // (the following lines are semi-automatically generated by that program)
-
-    // TODO: for a more independent test we compute J1(z) in some different way;
-    // see e.g. http://keisan.casio.com/exec/system/1180573474
+    // Test four arbitrary function values.
+    // Reference values are computed using dev-tools/math/cbesselJ01.c,
+    //   from which BesselJ1 has been derived. So this is _not_ an independent
+    //   for numeric accuracy, but only for invariance.
+    // However, the four specific results below have also been checked against the
+    //   online calculator http://keisan.casio.com/exec/system/1180573474.
+    //   Agreement is excellent, except for the dominantly imaginary argument .01+100i.
+    //   In conclusion, Bessel_J1 is clearly good enough for our purpose.
     res = MathFunctions::Bessel_J1(complex_t(0.8,1.5));
-    EXPECT_NEAR( res.real(), 0.72837687825769404, eps*std::abs(res) );
-    EXPECT_NEAR( res.imag(), 0.75030568686427268, eps*std::abs(res) );
+    EXPECT_NEAR( res.real(), 0.72837687825769404, eps*std::abs(res) ); // Keisan ..69398...
+    EXPECT_NEAR( res.imag(), 0.75030568686427268, eps*std::abs(res) ); // Keisan ..27264...
 
     res = MathFunctions::Bessel_J1(complex_t(1e-2,1e2));
-    EXPECT_NEAR( res.real(), 1.0630504683139779e+40, eps*std::abs(res) );
-    EXPECT_NEAR( res.imag(), 1.0683164984973165e+42, eps*std::abs(res) );
+    EXPECT_NEAR( res.real(), 1.0630504683139779e+40, eps*std::abs(res) ); // Keisan 1.063015...
+    EXPECT_NEAR( res.imag(), 1.0683164984973165e+42, eps*std::abs(res) ); // Keisan ..73162...
 
     res = MathFunctions::Bessel_J1(complex_t(-1e2,1e-2));
-    EXPECT_NEAR( res.real(), 0.077149198549289394, eps*std::abs(res) );
-    EXPECT_NEAR( res.imag(), 0.0002075766253119904, eps*std::abs(res) );
+    EXPECT_NEAR( res.real(), 0.077149198549289394, eps*std::abs(res) );  // Keisan ..89252...
+    EXPECT_NEAR( res.imag(), 2.075766253119904e-4, eps*std::abs(res) ); // Keisan ..19951...
 
     res = MathFunctions::Bessel_J1(complex_t(7,9));
-    EXPECT_NEAR( res.real(),       370.00180888861155, eps*std::abs(res) );
-    EXPECT_NEAR( res.imag(),       856.00300811057934, eps*std::abs(res) );
+    EXPECT_NEAR( res.real(), 370.00180888861155, eps*std::abs(res) ); // Keisan ..61107...
+    EXPECT_NEAR( res.imag(), 856.00300811057934, eps*std::abs(res) ); // Keisan ..57940...
 }
 
 // Test accuracy of complex function sinc(z) near the removable singularity at z=0