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125 lines
3.0 KiB
Plaintext
125 lines
3.0 KiB
Plaintext
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[section:bessel_over Bessel Function Overview]
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[h4 Ordinary Bessel Functions]
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Bessel Functions are solutions to Bessel's ordinary differential
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equation:
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[equation bessel1]
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where [nu][space] is the /order/ of the equation, and may be an arbitrary
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real or complex number, although integer orders are the most common occurrence.
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This library supports either integer or real orders.
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Since this is a second order differential equation, there must be two
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linearly independent solutions, the first of these is denoted J[sub v][space]
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and known as a Bessel function of the first kind:
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[equation bessel2]
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This function is implemented in this library as __cyl_bessel_j.
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The second solution is denoted either Y[sub v][space] or N[sub v][space]
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and is known as either a Bessel Function of the second kind, or as a
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Neumann function:
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[equation bessel3]
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This function is implemented in this library as __cyl_neumann.
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The Bessel functions satisfy the recurrence relations:
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[equation bessel4]
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[equation bessel5]
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Have the derivatives:
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[equation bessel6]
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[equation bessel7]
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Have the Wronskian relation:
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[equation bessel8]
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and the reflection formulae:
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[equation bessel9]
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[equation bessel10]
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[h4 Modified Bessel Functions]
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The Bessel functions are valid for complex argument /x/, and an important
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special case is the situation where /x/ is purely imaginary: giving a real
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valued result. In this case the functions are the two linearly
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independent solutions to the modified Bessel equation:
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[equation mbessel1]
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The solutions are known as the modified Bessel functions of the first and
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second kind (or occasionally as the hyperbolic Bessel functions of the first
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and second kind). They are denoted I[sub v][space] and K[sub v][space]
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respectively:
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[equation mbessel2]
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[equation mbessel3]
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These functions are implemented in this library as __cyl_bessel_i and
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__cyl_bessel_k respectively.
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The modified Bessel functions satisfy the recurrence relations:
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[equation mbessel4]
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[equation mbessel5]
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Have the derivatives:
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[equation mbessel6]
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[equation mbessel7]
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Have the Wronskian relation:
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[equation mbessel8]
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and the reflection formulae:
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[equation mbessel9]
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[equation mbessel10]
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[h4 Spherical Bessel Functions]
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When solving the Helmholtz equation in spherical coordinates by
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separation of variables, the radial equation has the form:
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[equation sbessel1]
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The two linearly independent solutions to this equation are called the
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spherical Bessel functions j[sub n][space] and y[sub n][space], and are related to the
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ordinary Bessel functions J[sub n][space] and Y[sub n][space] by:
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[equation sbessel2]
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The spherical Bessel function of the second kind y[sub n][space]
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is also known as the spherical Neumann function n[sub n].
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These functions are implemented in this library as __sph_bessel and
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__sph_neumann.
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[endsect]
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[/
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Copyright 2006 John Maddock, Paul A. Bristow and Xiaogang Zhang.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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