airy_bi

eve::airy_bi = functor<airy_bi_t>

inlineconstexpr

Header file

#include <eve/module/bessel.hpp>

Callable Signatures

namespace eve
{
   // Regular overload
   constexpr auto airy_bi(floating_value auto x) -> decltype(x)           noexcept; // 1
 
   // Lanes masking
   constexpr auto airy_bi[conditional_expr auto c](floating_value auto x) noexcept; // 2
   constexpr auto airy_bi[logical_value auto m](floating_value auto x)    noexcept; // 2

Parameters

• x: positive floating argument.
• c: Conditional expression masking the operation.
• m: Logical value masking the operation.

Return value

1. The value of \( \displaystyle Bi(x) = \frac1{\pi}\int_{0}^{\infty} \left[\exp\left(-{\frac{t^{3}}{3}}+xt\right)+ \sin\left({\frac{t^{3}}{3}}+xt\right)\,\right]dt\) is returned.
2. The operation is performed conditionnaly..

External references

• Wolfram MathWorld: Airy Functions
• Wikipedia: Airy function
• DLMF: Airy and Related Functions

Example

// revision 1
#include <eve/module/bessel.hpp>
#include <iostream>
 
eve::wide<double> wf0([](auto i, auto c)->double{ return 2*(i-c/2);});
eve::wide         wf1{0.5, -1.5, 0.1, -1.0, 19.0, 25.0, 21.5, 10000.0};
 
int main()
{
   std::cout << "<- wf0 = " << wf0 << "\n";
   std::cout << "<- wf1 = " << wf1 << "\n";
 
   std::cout << "-> biry_bi(wf0)                = " << eve::airy_bi(wf0) << "\n";
   std::cout << "-> airy_bi[ignore_last(2)](wf0)= " << eve::airy_bi[eve::ignore_last(2)](wf0) << "\n";
   std::cout << "-> airy_bi[wf0 != -2.0f](wf0)  = " << eve::airy_bi[wf0 != -2.0f](wf0) << "\n";
   std::cout << "-> airy_bi(wf1)                = " << eve::airy_bi(wf1) << "\n";
 
}

It is the solution of the differential equation \(y''-xy = 0\) satisfying \(Ai(0) = \frac1{3^{1/6}\Gamma(2/3)}\) and \(Ai'(0) =\frac{3^{1/6}}{\Gamma(1/3)}\).