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/*!
* \file
* \brief Definitions of Bessel functions
* \author Tony Ottosson
*
* -------------------------------------------------------------------------
*
* IT++ - C++ library of mathematical, signal processing, speech processing,
* and communications classes and functions
*
* Copyright (C) 1995-2008 (see AUTHORS file for a list of contributors)
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
*
* -------------------------------------------------------------------------
*/
#ifndef BESSEL_H
#define BESSEL_H
#include <itpp/base/vec.h>
namespace itpp {
/*! \addtogroup besselfunctions
*/
/*!
\ingroup besselfunctions
\brief Bessel function of first kind of order \a nu for \a nu integer
The bessel function of first kind is defined as:
\f[
J_{\nu}(x) = \sum_{k=0}^{\infty} \frac{ (-1)^{k} }{k! \Gamma(\nu+k+1) } \left(\frac{x}{2}\right)^{\nu+2k}
\f]
where \f$\nu\f$ is the order and \f$ 0 < x < \infty \f$.
*/
double besselj(int nu, double x);
/*!
\ingroup besselfunctions
\brief Bessel function of first kind of order \a nu for \a nu integer
*/
vec besselj(int nu, const vec &x);
/*!
\ingroup besselfunctions
\brief Bessel function of first kind of order \a nu. \a nu is real.
*/
double besselj(double nu, double x);
/*!
\ingroup besselfunctions
\brief Bessel function of first kind of order \a nu. \a nu is real.
*/
vec besselj(double nu, const vec &x);
/*!
\ingroup besselfunctions
\brief Bessel function of second kind of order \a nu. \a nu is integer.
The Bessel function of second kind is defined as:
\f[
Y_{\nu}(x) = \frac{J_{\nu}(x) \cos(\nu\pi) - J_{-\nu}(x)}{\sin(\nu\pi)}
\f]
where \f$\nu\f$ is the order and \f$ 0 < x < \infty \f$.
*/
double bessely(int nu, double x);
/*!
\ingroup besselfunctions
\brief Bessel function of second kind of order \a nu. \a nu is integer.
*/
vec bessely(int nu, const vec &x);
/*!
\ingroup besselfunctions
\brief Bessel function of second kind of order \a nu. \a nu is real.
*/
double bessely(double nu, double x);
/*!
\ingroup besselfunctions
\brief Bessel function of second kind of order \a nu. \a nu is real.
*/
vec bessely(double nu, const vec &x);
/*!
\ingroup besselfunctions
\brief Modified Bessel function of first kind of order \a nu. \a nu is \a double. \a x is \a double.
The Modified Bessel function of first kind is defined as:
\f[
I_{\nu}(x) = i^{-\nu} J_{\nu}(ix)
\f]
where \f$\nu\f$ is the order and \f$ 0 < x < \infty \f$.
*/
double besseli(double nu, double x);
/*!
\ingroup besselfunctions
\brief Modified Bessel function of first kind of order \a nu. \a nu is \a double. \a x is \a double.
*/
vec besseli(double nu, const vec &x);
/*!
\ingroup besselfunctions
\brief Modified Bessel function of second kind of order \a nu. \a nu is double. \a x is double.
The Modified Bessel function of second kind is defined as:
\f[
K_{\nu}(x) = \frac{\pi}{2} i^{\nu+1} [J_{\nu}(ix) + i Y_{\nu}(ix)]
\f]
where \f$\nu\f$ is the order and \f$ 0 < x < \infty \f$.
*/
double besselk(int nu, double x);
/*!
\ingroup besselfunctions
\brief Modified Bessel function of second kind of order \a nu. \a nu is double. \a x is double.
*/
vec besselk(int nu, const vec &x);
} //namespace itpp
#endif // #ifndef BESSEL_H
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