Definition:Hankel Function
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Definition
A Hankel function is the sum of Bessel functions in either of the following two ways:
Hankel Function of the First Kind
The Hankel function of the first kind is defined as:
- $\map {H_n^{\paren 1} } z = \map {J_n} z + i \map {Y_n} z$
where:
- $\map {J_n} z$ denotes the Bessel function of the first kind of order $n$
- $\map {Y_n} z$ denotes the Bessel function of the second kind of order $n$.
Hankel Function of the Second Kind
The Hankel function of the second kind is defined as:
- $\map {H_n^{\paren 2} } z = \map {J_n} z - i \map {Y_n} z$
where:
- $\map {J_n} z$ denotes the Bessel function of the first kind of order $n$
- $\map {Y_n} z$ denotes the Bessel function of the second kind of order $n$.
Also known as
The Hankel functions are also known as the Bessel functions of the third kind.
However, as there are two kinds of Hankel functions:
referring to them as Bessel functions of the third kind is a recipe for confusion.
Also see
- Results about Hankel functions can be found here.
Source of Name
This entry was named for Hermann Hankel.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Bessel functions
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Hankel function
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Bessel functions
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hankel function