Bessel function

Detailed Description

Bessel functions.

This module provides implementation for the Bessel functions and related operations.

Required header:

#include <eve/module/bessel.hpp> 

Variables
constexpr auto	eve::airy = functor<airy_t>
	elementwise_callable object computing simultaneously the airy functions values \( Ai(x)\) and \( Bi(x)\).
constexpr auto	eve::airy_ai = functor<airy_ai_t>
	elementwise_callable object computing the airy function \( Ai(x)\).
constexpr auto	eve::airy_bi = functor<airy_bi_t>
	elementwise_callable object computing the airy function \( Bi(x)\).

Variable Documentation

airy

auto eve::airy = functor<airy_t>

inlineconstexpr

elementwise_callable object computing simultaneously the airy functions values \( Ai(x)\) and \( Bi(x)\).

This function is designed to be faster than two separate calls to eve::airy_ai and eve::airy_bi.

Header file

#include <eve/module/bessel.hpp>

Callable Signatures

namespace eve
{
   constexpr auto airy(eve::floating_value auto x) -> decltype(eve::zip(x,x)) noexcept;
}

Parameters

• x : floating value.

Return value

The tuple {airy_ai (x), airy_bi (x)} is returned.

External references

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

Example

#include <eve/eve.hpp>
#include <eve/module/bessel.hpp>
#include <iostream>
 
int main()
{
   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};
 
   std::cout << "<- wf0 = " << wf0 << "\n";
   std::cout << "<- wf1 = " << wf1 << "\n";
 
   std::cout << "-> airy(wf0) = " << eve::airy(wf0) << "\n";
   std::cout << "-> airy(wf1) = " << eve::airy(wf1) << "\n";
   std::cout << eve::airy(2.0) << std::endl;
}

airy_ai

auto eve::airy_ai = functor<airy_ai_t>

inlineconstexpr

elementwise_callable object computing the airy function \( Ai(x)\).

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

Header file

#include <eve/module/bessel.hpp>

Callable Signatures

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

Parameters

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

Return value

1. The value of \( \displaystyle Ai(x)=\frac1{\pi} \int_{0}^{\infty}\cos \left( \frac{t^3}{3}+xt \right) \,dt\) is returned.
2. The operation is performed conditionally..

External references

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

Example

#include <eve/eve.hpp>
#include <eve/module/bessel.hpp>
#include <iostream>
 
int main()
{
   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};
 
   std::cout << "<- wf0 = " << wf0 << "\n";
   std::cout << "<- wf1 = " << wf1 << "\n";
 
   std::cout << "-> airy_ai(wf0)                = " << eve::airy_ai(wf0) << "\n";
   std::cout << "-> airy_ai[ignore_last(2)](wf0)= " << eve::airy_ai[eve::ignore_last(2)](wf0) << "\n";
   std::cout << "-> airy_ai[wf0 != -2.0f](wf0)  = " << eve::airy_ai[wf0 != -2.0f](wf0) << "\n";
   std::cout << "-> airy_ai(wf1)                = " << eve::airy_ai(wf1) << "\n";
}

airy_bi

auto eve::airy_bi = functor<airy_bi_t>

inlineconstexpr

elementwise_callable object computing the airy function \( Bi(x)\).

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

Header file

#include <eve/module/bessel.hpp>

Callable Signatures

namespace eve
{
   // Regular overload
   constexpr auto airy_bi(floating_value auto 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 conditionally..

External references

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

Example

#include <eve/eve.hpp>
#include <eve/module/bessel.hpp>
#include <iostream>
 
int main()
{
   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};
 
   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";
}