Bessel function: Difference between revisions
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'''Bessel functions''', in mathematics, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions ''y''(''x'') of Bessel's differential equation: | '''Bessel functions''', in mathematics, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions ''y''(''x'') of Bessel's differential equation: | ||
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Although α and −α produce the same differential equation, it is conventional to define different Bessel functions for these two orders (e.g., so that the Bessel functions are mostly smooth functions of α). Bessel functions are also known as '''Cylinder functions''' or '''Cylindrical harmonics''' because they are found in the solution to Laplace's equation in cylindrical coordinates. | Although α and −α produce the same differential equation, it is conventional to define different Bessel functions for these two orders (e.g., so that the Bessel functions are mostly smooth functions of α). Bessel functions are also known as '''Cylinder functions''' or '''Cylindrical harmonics''' because they are found in the solution to Laplace's equation in cylindrical coordinates. | ||
The [[ | ==Usage== | ||
The [[Ascendant Justice's AI|artificial intelligence]] aboard the ''[[Ascendant Justice]]'' attempted to use Bessel functions to communicate with [[Cortana]].<ref>''[[Halo: First Strike]]'', Chapter 7</ref> [[Dominique]] had a tattoo of a Bessel function on his left wrist.<ref>''[[Halo: The Fall of Reach]]'', Chapter 29</ref> | |||
==List of appearances== | |||
*''[[Halo: The Fall of Reach]]'' {{1st}} | |||
*''[[Halo: First Strike]]'' | |||
==Sources== | ==Sources== | ||
{{Ref/Sources}} | |||
[[Category:Science]] | [[Category:Science]] | ||
Latest revision as of 12:43, March 23, 2022
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There is more information available on this subject at Bessel function on the English Wikipedia. |
Bessel functions, in mathematics, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation:
- x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0
For an arbitrary real or complex number α. The most common and important special case is where α is an integer n, then α is referred to as the order of the Bessel function.
Although α and −α produce the same differential equation, it is conventional to define different Bessel functions for these two orders (e.g., so that the Bessel functions are mostly smooth functions of α). Bessel functions are also known as Cylinder functions or Cylindrical harmonics because they are found in the solution to Laplace's equation in cylindrical coordinates.
Usage[edit]
The artificial intelligence aboard the Ascendant Justice attempted to use Bessel functions to communicate with Cortana.[1] Dominique had a tattoo of a Bessel function on his left wrist.[2]
List of appearances[edit]
- Halo: The Fall of Reach (First appearance)
- Halo: First Strike
Sources[edit]
- ^ Halo: First Strike, Chapter 7
- ^ Halo: The Fall of Reach, Chapter 29
