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Bessel differential equation
[edit]The solutions of the Bessel differential equation[1]
| x 2 + x + (x 2 − ν 2 ) y = 0 |
are the Bessel functions,[2] of which there are two kinds:
- Bessel functions of the first kind [3]: nonsingular at the origin;
- Bessel functions of the second kind [4]: singular at the origin.
Bessel functions of the first kind
[edit]
The Bessel functions of the first kind
...
Bessel functions of the first kind (integer order)
[edit]
The Bessel functions of the first kind
, with nonnegative order
are also known as cylindrical Bessel functions.
Zeros of Bessel functions of the first kind (integer order):[5]
- For the decimal expansion of first zero of the Bessel functions
| J0(z), J1(z), J2(z), J3(z), J4(z), J5(z), |
see: A115368, A115369, A115370, A115371, A115372, A115373.
- For the decimal expansion of second zero of the Bessel functions
| J0(z), J1(z), J2(z), J3(z), J4(z), J5(z), |
see: A280868 , A??????, A??????, A??????, A??????, A??????.
Bessel functions of the first kind (half-integer order)
[edit](...)
Bessel functions of the second kind
[edit]
The Bessel functions of the second kind
...
Bessel functions of the second kind (integer order)
[edit]
The Bessel functions of the second kind
, with nonnegative order
Bessel functions of the second kind (half-integer order)
[edit](...)
Modified Bessel differential equation
[edit]The solutions of the modified Bessel differential equation[6]
| x 2 + x − (x 2 − ν 2 ) y = 0 |
are the modified Bessel functions, of which there are two kinds:
- Modified Bessel functions of the first kind [7]: nonsingular at the origin;
- Modified Bessel functions of the second kind [8]: singular at the origin.
Spherical Bessel differential equation
[edit]The solutions of the spherical Bessel differential equation[9]
| x 2 + 2 x + [x 2 − ν (ν + 1)] y = 0 |
are the spherical Bessel functions,[10] of which there are two kinds:
- Spherical Bessel functions of the first kind [11]: nonsingular at the origin;
- Spherical Bessel functions of the second kind [12]: singular at the origin.
Modified spherical Bessel differential equation
[edit]The solutions of the modified spherical Bessel differential equation[13]
| x 2 + 2 x − [x 2 − ν (ν + 1)] y = 0 |
are the modified spherical Bessel functions,[14] of which there are two kinds:
- Modified spherical Bessel functions of the first kind [15]: nonsingular at the origin;
- Modified spherical Bessel functions of the second kind [16]: singular at the origin.
- ↑ Weisstein, Eric W., Bessel Differential Equation, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Bessel Function, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Bessel Function of the First Kind, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Bessel Function of the Second Kind, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Bessel Function Zeros, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Modified Bessel Differential Equation, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Modified Bessel Function of the First Kind, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Modified Bessel Function of the Second Kind, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Spherical Bessel Differential Equation, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Spherical Bessel Function, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Spherical Bessel Function of the First Kind, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Spherical Bessel Function of the Second Kind, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Modified Spherical Bessel Differential Equation, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Modified Spherical Bessel Function, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Modified Spherical Bessel Function of the First Kind, from MathWorld—A Wolfram Web Resource.
- ↑ Weisstein, Eric W., Modified Spherical Bessel Function of the Second Kind, from MathWorld—A Wolfram Web Resource.
