/*!
 * \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) 1995-2007  (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