Bessel functions

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Type

First kind

Second kind

Bessel functions

Jν

Yν

Modified Bessel functions

I ν

K ν

Hankel functions

H (1)ν = Jν + i Yν

H (2)ν = Jν i Yν

Spherical Bessel functions

jν

yν

Modified spherical Bessel functions

i ν

k ν

Spherical Hankel functions

h (1)ν = jν + i yν

h (2)ν = jν i yν

Bessel differential equation

The solutions of the Bessel differential equation^[1]

x 2

d 2 y

d x 2

+ x

d y

d x

+ (x 2 − ν 2 ) y = 0

are the Bessel functions,^[2] of which there are two kinds:

Bessel functions of the first kind
Jν (x)
^[3]: nonsingular at the origin;
Bessel functions of the second kind
Yν (x)
^[4]: singular at the origin.

1 Bessel differential equation
- 1.1 Bessel functions of the first kind
  - 1.1.1 Bessel functions of the first kind (integer order)
  - 1.1.2 Bessel functions of the first kind (half-integer order)
- 1.2 Bessel functions of the second kind
  - 1.2.1 Bessel functions of the second kind (integer order)
  - 1.2.2 Bessel functions of the second kind (half-integer order)
2 Modified Bessel differential equation
3 Spherical Bessel differential equation
4 Modified spherical Bessel differential equation
5 Notes
6 External links

Bessel functions of the first kind

The Bessel functions of the first kind

Jν (x)

...

Bessel functions of the first kind (integer order)

The Bessel functions of the first kind

Jn (x)

, with nonnegative order

n = 0, 1, 2, 3, ...

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)

(...)

Bessel functions of the second kind

The Bessel functions of the second kind

Yν (x)

...

Bessel functions of the second kind (integer order)

The Bessel functions of the second kind

Yn (x)

, with nonnegative order

n = 0, 1, 2, 3, ...

Bessel functions of the second kind (half-integer order)

(...)

Modified Bessel differential equation

The solutions of the modified Bessel differential equation^[6]

x 2

d 2 y

d x 2

+ x

d y

d 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
I ν (x)
^[7]: nonsingular at the origin;
Modified Bessel functions of the second kind
K ν (x)
^[8]: singular at the origin.

Spherical Bessel differential equation

The solutions of the spherical Bessel differential equation^[9]

x 2

d 2 y

d x 2

+ 2 x

d y

d 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
jν (x)
^[11]: nonsingular at the origin;
Spherical Bessel functions of the second kind
yν (x)
^[12]: singular at the origin.

Modified spherical Bessel differential equation

The solutions of the modified spherical Bessel differential equation^[13]

x 2

d 2 y

d x 2

+ 2 x

d y

d 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
i ν (x)
^[15]: nonsingular at the origin;
Modified spherical Bessel functions of the second kind
k ν (x)
^[16]: singular at the origin.

Notes

↑ 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.

External links

J.R. Culham, Bessel Functions of the First and Second Kind.
Martin Kreh, The Bessel Functions.

[1] Weisstein, Eric W., Bessel Differential Equation, from MathWorld—A Wolfram Web Resource.

[2] Weisstein, Eric W., Bessel Function, from MathWorld—A Wolfram Web Resource.

[3] Weisstein, Eric W., Bessel Function of the First Kind, from MathWorld—A Wolfram Web Resource.

[4] Weisstein, Eric W., Bessel Function of the Second Kind, from MathWorld—A Wolfram Web Resource.

[5] Weisstein, Eric W., Bessel Function Zeros, from MathWorld—A Wolfram Web Resource.

[6] Weisstein, Eric W., Modified Bessel Differential Equation, from MathWorld—A Wolfram Web Resource.

[7] Weisstein, Eric W., Modified Bessel Function of the First Kind, from MathWorld—A Wolfram Web Resource.

[8] Weisstein, Eric W., Modified Bessel Function of the Second Kind, from MathWorld—A Wolfram Web Resource.

[9] Weisstein, Eric W., Spherical Bessel Differential Equation, from MathWorld—A Wolfram Web Resource.

[10] Weisstein, Eric W., Spherical Bessel Function, from MathWorld—A Wolfram Web Resource.

[11] Weisstein, Eric W., Spherical Bessel Function of the First Kind, from MathWorld—A Wolfram Web Resource.

[12] Weisstein, Eric W., Spherical Bessel Function of the Second Kind, from MathWorld—A Wolfram Web Resource.

[13] Weisstein, Eric W., Modified Spherical Bessel Differential Equation, from MathWorld—A Wolfram Web Resource.

[14] Weisstein, Eric W., Modified Spherical Bessel Function, from MathWorld—A Wolfram Web Resource.

[15] Weisstein, Eric W., Modified Spherical Bessel Function of the First Kind, from MathWorld—A Wolfram Web Resource.

[16] Weisstein, Eric W., Modified Spherical Bessel Function of the Second Kind, from MathWorld—A Wolfram Web Resource.

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Bessel functions

Bessel differential equation

Contents

Bessel functions of the first kind

Bessel functions of the first kind (integer order)

Bessel functions of the first kind (half-integer order)

Bessel functions of the second kind

Bessel functions of the second kind (integer order)

Bessel functions of the second kind (half-integer order)

Modified Bessel differential equation

Spherical Bessel differential equation

Modified spherical Bessel differential equation

Notes

External links

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