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Bessel Functions

Functions

double itpp::besselj (int nu, double x)
 Bessel function of first kind of order nu for nu integer.
 
vec itpp::besselj (int nu, const vec &x)
 Bessel function of first kind of order nu for nu integer.
 
double itpp::besselj (double nu, double x)
 Bessel function of first kind of order nu. nu is real.
 
vec itpp::besselj (double nu, const vec &x)
 Bessel function of first kind of order nu. nu is real.
 
double itpp::bessely (int nu, double x)
 Bessel function of second kind of order nu. nu is integer.
 
vec itpp::bessely (int nu, const vec &x)
 Bessel function of second kind of order nu. nu is integer.
 
double itpp::bessely (double nu, double x)
 Bessel function of second kind of order nu. nu is real.
 
vec itpp::bessely (double nu, const vec &x)
 Bessel function of second kind of order nu. nu is real.
 
double itpp::besseli (double nu, double x)
 Modified Bessel function of first kind of order nu. nu is double. x is double.
 
vec itpp::besseli (double nu, const vec &x)
 Modified Bessel function of first kind of order nu. nu is double. x is double.
 
double itpp::besselk (int nu, double x)
 Modified Bessel function of second kind of order nu. nu is double. x is double.
 
vec itpp::besselk (int nu, const vec &x)
 Modified Bessel function of second kind of order nu. nu is double. x is double.
 

Detailed Description

end of math group

Function Documentation

◆ besselj() [1/4]

ITPP_EXPORT double itpp::besselj ( int nu,
double x )

Bessel function of first kind of order nu for nu integer.

The bessel function of first kind is defined as:

\[
J_{\nu}(x) = \sum_{k=0}^{\infty} \frac{ (-1)^{k} }{k! \Gamma(\nu+k+1) } \left(\frac{x}{2}\right)^{\nu+2k}
\]

where $\nu$ is the order and $ 0 < x < \infty $.

Definition at line 44 of file bessel.cpp.

Referenced by itpp::FIR_Fading_Generator::Jakes_filter().

◆ besselj() [2/4]

ITPP_EXPORT vec itpp::besselj ( int nu,
const vec & x )

Bessel function of first kind of order nu for nu integer.

Definition at line 46 of file bessel.cpp.

◆ besselj() [3/4]

ITPP_EXPORT double itpp::besselj ( double nu,
double x )

Bessel function of first kind of order nu. nu is real.

Definition at line 56 of file bessel.cpp.

◆ besselj() [4/4]

ITPP_EXPORT vec itpp::besselj ( double nu,
const vec & x )

Bessel function of first kind of order nu. nu is real.

Definition at line 58 of file bessel.cpp.

◆ bessely() [1/4]

ITPP_EXPORT double itpp::bessely ( int nu,
double x )

Bessel function of second kind of order nu. nu is integer.

The Bessel function of second kind is defined as:

\[
Y_{\nu}(x) = \frac{J_{\nu}(x) \cos(\nu\pi) - J_{-\nu}(x)}{\sin(\nu\pi)}
\]

where $\nu$ is the order and $ 0 < x < \infty $.

Definition at line 68 of file bessel.cpp.

◆ bessely() [2/4]

ITPP_EXPORT vec itpp::bessely ( int nu,
const vec & x )

Bessel function of second kind of order nu. nu is integer.

Definition at line 70 of file bessel.cpp.

◆ bessely() [3/4]

ITPP_EXPORT double itpp::bessely ( double nu,
double x )

Bessel function of second kind of order nu. nu is real.

Definition at line 79 of file bessel.cpp.

◆ bessely() [4/4]

ITPP_EXPORT vec itpp::bessely ( double nu,
const vec & x )

Bessel function of second kind of order nu. nu is real.

Definition at line 81 of file bessel.cpp.

◆ besseli() [1/2]

ITPP_EXPORT double itpp::besseli ( double nu,
double x )

Modified Bessel function of first kind of order nu. nu is double. x is double.

The Modified Bessel function of first kind is defined as:

\[
I_{\nu}(x) = i^{-\nu} J_{\nu}(ix)
\]

where $\nu$ is the order and $ 0 < x < \infty $.

Definition at line 91 of file bessel.cpp.

◆ besseli() [2/2]

ITPP_EXPORT vec itpp::besseli ( double nu,
const vec & x )

Modified Bessel function of first kind of order nu. nu is double. x is double.

Definition at line 93 of file bessel.cpp.

◆ besselk() [1/2]

ITPP_EXPORT double itpp::besselk ( int nu,
double x )

Modified Bessel function of second kind of order nu. nu is double. x is double.

The Modified Bessel function of second kind is defined as:

\[
K_{\nu}(x) = \frac{\pi}{2} i^{\nu+1} [J_{\nu}(ix) + i Y_{\nu}(ix)]
\]

where $\nu$ is the order and $ 0 < x < \infty $.

Definition at line 103 of file bessel.cpp.

◆ besselk() [2/2]

ITPP_EXPORT vec itpp::besselk ( int nu,
const vec & x )

Modified Bessel function of second kind of order nu. nu is double. x is double.

Definition at line 105 of file bessel.cpp.

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