...one of the most highly
regarded and expertly designed C++ library projects in the
world.
— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards
#include <boost/math/special_functions/bessel.hpp>
template <class T1, class T2> calculatedresulttype sph_bessel(unsigned v, T2 x); template <class T1, class T2, class Policy> calculatedresulttype sph_bessel(unsigned v, T2 x, const Policy&); template <class T1, class T2> calculatedresulttype sph_neumann(unsigned v, T2 x); template <class T1, class T2, class Policy> calculatedresulttype sph_neumann(unsigned v, T2 x, const Policy&);
The functions sph_bessel and sph_neumann return the result of the Spherical Bessel functions of the first and second kinds respectively:
sph_bessel(v, x) = j_{v}(x)
sph_neumann(v, x) = y_{v}(x) = n_{v}(x)
where:
The return type of these functions is computed using the result type calculation rules for the single argument type T.
The final Policy argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the policy documentation for more details.
The functions return the result of domain_error
whenever the result is undefined or complex: this occurs when x < 0
.
The j_{v} function is cyclic like J_{v} but differs in its behaviour at the origin:
Likewise y_{v} is also cyclic for large x, but tends to ∞ for small x:
There are two sets of test values: spot values calculated using functions.wolfram.com, and a much larger set of tests computed using a simplified version of this implementation (with all the special case handling removed).
Table 7.48. Error rates for sph_bessel
GNU C++ version 7.1.0 
GNU C++ version 7.1.0 
Sun compiler version 0x5150 
Microsoft Visual C++ version 14.1 


Bessel j: Random Data 
Max = 243ε (Mean = 13.3ε) 
Max = 0.978ε (Mean = 0.0445ε) 
Max = 243ε (Mean = 33.7ε) 
Max = 245ε (Mean = 16.3ε) 
Table 7.49. Error rates for sph_neumann
GNU C++ version 7.1.0 
GNU C++ version 7.1.0 
Sun compiler version 0x5150 
Microsoft Visual C++ version 14.1 


y: Random Data 
Max = 234ε (Mean = 19.5ε) 
Max = 0.995ε (Mean = 0.0665ε) 
Max = 234ε (Mean = 19.8ε) 
Max = 281ε (Mean = 31.1ε) 
Other than error handling and a couple of special cases these functions are implemented directly in terms of their definitions:
The special cases occur for:
j_{0} = sinc_pi(x) = sin(x) / x
and for small x < 1, we can use the series:
which neatly avoids the problem of calculating 0/0 that can occur with the main definition as x → 0.