ROOT 6.08/07 Reference Guide 
This class calculates the Kelvin functions Ber(x), Bei(x), Ker(x), Kei(x), and their first derivatives.
Definition at line 38 of file KelvinFunctions.h.
Public Member Functions  
virtual  ~KelvinFunctions () 
Static Public Member Functions  
static double  Bei (double x) 
\[ Bei(x) = Bei_{0}(x) = Im\left[J_{0}\left(x e^{3\pi i/4}\right)\right] \] where x is real, and \(J_{0}(z)\) is the zerothorder Bessel function of the first kind. More...  
static double  Ber (double x) 
\[ Ber(x) = Ber_{0}(x) = Re\left[J_{0}\left(x e^{3\pi i/4}\right)\right] \] where x is real, and \(J_{0}(z)\) is the zerothorder Bessel function of the first kind. More...  
static double  DBei (double x) 
Calculates the first derivative of Bei(x). More...  
static double  DBer (double x) 
Calculates the first derivative of Ber(x). More...  
static double  DKei (double x) 
Calculates the first derivative of Kei(x). More...  
static double  DKer (double x) 
Calculates the first derivative of Ker(x). More...  
static double  F1 (double x) 
Utility function appearing in the calculations of the Kelvin functions Bei(x) and Ber(x) (and their derivatives). More...  
static double  F2 (double x) 
Utility function appearing in the calculations of the Kelvin functions Kei(x) and Ker(x) (and their derivatives). More...  
static double  G1 (double x) 
Utility function appearing in the calculations of the Kelvin functions Bei(x) and Ber(x) (and their derivatives). More...  
static double  G2 (double x) 
Utility function appearing in the calculations of the Kelvin functions Kei(x) and Ker(x) (and their derivatives). More...  
static double  Kei (double x) 
\[ Kei(x) = Kei_{0}(x) = Im\left[K_{0}\left(x e^{3\pi i/4}\right)\right] \] where x is real, and \(K_{0}(z)\) is the zerothorder modified Bessel function of the second kind. More...  
static double  Ker (double x) 
\[ Ker(x) = Ker_{0}(x) = Re\left[K_{0}\left(x e^{3\pi i/4}\right)\right] \] where x is real, and \(K_{0}(z)\) is the zerothorder modified Bessel function of the second kind. More...  
static double  M (double x) 
Utility function appearing in the asymptotic expansions of DBer(x) and DBei(x). More...  
static double  N (double x) 
Utility function appearing in the asymptotic expansions of DKer(x) and DKei(x). More...  
static double  Phi (double x) 
Utility function appearing in the asymptotic expansions of DKer(x) and DKei(x). More...  
static double  Theta (double x) 
Utility function appearing in the asymptotic expansions of DBer(x) and DBei(x). More...  
Static Protected Attributes  
static double  fgEpsilon = 1.e20 
static double  fgMin = 20 
#include <Math/KelvinFunctions.h>

inlinevirtual 
Definition at line 63 of file KelvinFunctions.h.

static 
\[ Bei(x) = Bei_{0}(x) = Im\left[J_{0}\left(x e^{3\pi i/4}\right)\right] \]
where x is real, and \(J_{0}(z)\) is the zerothorder Bessel function of the first kind.
If x < fgMin (=20), Bei(x) is computed according to its polynomial approximation
\[ Bei(x) = \sum_{n \geq 0}\frac{(1)^{n}(x/2)^{4n+2}}{[(2n+1)!]^{2}} \]
For x > fgMin, Bei(x) is computed according to its asymptotic expansion:
\[ Bei(x) = \frac{e^{x/\sqrt{2}}}{\sqrt{2\pi x}} [F1(x) sin\alpha + G1(x) cos\alpha]  \frac{1}{\pi}Ker(x) \]
where \(\alpha = \frac{x}{\sqrt{2}}  \frac{\pi}{8}\).
Definition at line 130 of file KelvinFunctions.cxx.

static 
\[ Ber(x) = Ber_{0}(x) = Re\left[J_{0}\left(x e^{3\pi i/4}\right)\right] \]
where x is real, and \(J_{0}(z)\) is the zerothorder Bessel function of the first kind.
If x < fgMin (=20), Ber(x) is computed according to its polynomial approximation
\[ Ber(x) = 1 + \sum_{n \geq 1}\frac{(1)^{n}(x/2)^{4n}}{[(2n)!]^{2}} \]
For x > fgMin, Ber(x) is computed according to its asymptotic expansion:
\[ Ber(x) = \frac{e^{x/\sqrt{2}}}{\sqrt{2\pi x}} [F1(x) cos\alpha + G1(x) sin\alpha]  \frac{1}{\pi}Kei(x) \]
where \(\alpha = \frac{x}{\sqrt{2}}  \frac{\pi}{8}\).
Definition at line 72 of file KelvinFunctions.cxx.

static 
Calculates the first derivative of Bei(x).
If x < fgMin (=20), DBei(x) is computed according to the derivative of the polynomial approximation of Bei(x). Otherwise it is computed according to its asymptotic expansion
\[ \frac{d}{dx} Bei(x) = M sin\left(\theta  \frac{\pi}{4}\right) \]
Definition at line 359 of file KelvinFunctions.cxx.

static 
Calculates the first derivative of Ber(x).
If x < fgMin (=20), DBer(x) is computed according to the derivative of the polynomial approximation of Ber(x). Otherwise it is computed according to its asymptotic expansion
\[ \frac{d}{dx} Ber(x) = M cos\left(\theta  \frac{\pi}{4}\right) \]
Definition at line 314 of file KelvinFunctions.cxx.

static 
Calculates the first derivative of Kei(x).
If x < fgMin (=20), DKei(x) is computed according to the derivative of the polynomial approximation of Kei(x). Otherwise it is computed according to its asymptotic expansion
\[ \frac{d}{dx} Kei(x) = N sin\left(\phi  \frac{\pi}{4}\right) \]
Definition at line 452 of file KelvinFunctions.cxx.

static 
Calculates the first derivative of Ker(x).
If x < fgMin (=20), DKer(x) is computed according to the derivative of the polynomial approximation of Ker(x). Otherwise it is computed according to its asymptotic expansion
\[ \frac{d}{dx} Ker(x) = N cos\left(\phi  \frac{\pi}{4}\right) \]
Definition at line 404 of file KelvinFunctions.cxx.

static 
Utility function appearing in the calculations of the Kelvin functions Bei(x) and Ber(x) (and their derivatives).
F1(x) is given by
\[ F1(x) = 1 + \sum_{n \geq 1} \frac{\prod_{m=1}^{n}(2m  1)^{2}}{n! (8x)^{n}} cos\left(\frac{n\pi}{4}\right) \]
Definition at line 487 of file KelvinFunctions.cxx.

static 
Utility function appearing in the calculations of the Kelvin functions Kei(x) and Ker(x) (and their derivatives).
F2(x) is given by
\[ F2(x) = 1 + \sum_{n \geq 1} (1)^{n} \frac{\prod_{m=1}^{n}(2m  1)^{2}}{n! (8x)^{n}} cos\left(\frac{n\pi}{4}\right) \]
Definition at line 516 of file KelvinFunctions.cxx.

static 
Utility function appearing in the calculations of the Kelvin functions Bei(x) and Ber(x) (and their derivatives).
G1(x) is given by
\[ G1(x) = \sum_{n \geq 1} \frac{\prod_{m=1}^{n}(2m  1)^{2}}{n! (8x)^{n}} sin\left(\frac{n\pi}{4}\right) \]
Definition at line 547 of file KelvinFunctions.cxx.

static 
Utility function appearing in the calculations of the Kelvin functions Kei(x) and Ker(x) (and their derivatives).
G2(x) is given by
\[ G2(x) = \sum_{n \geq 1} (1)^{n} \frac{\prod_{m=1}^{n}(2m  1)^{2}}{n! (8x)^{n}} sin\left(\frac{n\pi}{4}\right) \]
Definition at line 574 of file KelvinFunctions.cxx.

static 
\[ Kei(x) = Kei_{0}(x) = Im\left[K_{0}\left(x e^{3\pi i/4}\right)\right] \]
where x is real, and \(K_{0}(z)\) is the zerothorder modified Bessel function of the second kind.
If x < fgMin (=20), Kei(x) is computed according to its polynomial approximation
\[ Kei(x) = \left(ln \frac{x}{2} + \gamma\right) Bei(x)  \left(\frac{\pi}{4}  \delta\right) Ber(x) + \sum_{n \geq 0} \frac{(1)^{n}}{[(2n)!]^{2}} H_{2n} \left(\frac{x}{2}\right)^{4n+2} \]
where \(\gamma = 0.577215664...\) is the EulerMascheroni constant, \(\delta = \pi\) for x < 0 and is otherwise zero, and
\[ H_{n} = \sum_{k = 1}^{n} \frac{1}{k} \]
For x > fgMin, Kei(x) is computed according to its asymptotic expansion:
\[ Kei(x) =  \sqrt{\frac{\pi}{2x}} e^{x/\sqrt{2}} [F2(x) sin\beta + G2(x) cos\beta] \]
where \(\beta = \frac{x}{\sqrt{2}} + \frac{\pi}{8}\).
Definition at line 262 of file KelvinFunctions.cxx.

static 
\[ Ker(x) = Ker_{0}(x) = Re\left[K_{0}\left(x e^{3\pi i/4}\right)\right] \]
where x is real, and \(K_{0}(z)\) is the zerothorder modified Bessel function of the second kind.
If x < fgMin (=20), Ker(x) is computed according to its polynomial approximation
\[ Ker(x) = \left(ln \frac{x}{2} + \gamma\right) Ber(x) + \left(\frac{\pi}{4}  \delta\right) Bei(x) + \sum_{n \geq 0} \frac{(1)^{n}}{[(2n)!]^{2}} H_{2n} \left(\frac{x}{2}\right)^{4n} \]
where \(\gamma = 0.577215664...\) is the EulerMascheroni constant, \(\delta = \pi\) for x < 0 and is otherwise zero, and
\[ H_{n} = \sum_{k = 1}^{n} \frac{1}{k} \]
For x > fgMin, Ker(x) is computed according to its asymptotic expansion:
\[ Ker(x) = \sqrt{\frac{\pi}{2x}} e^{x/\sqrt{2}} [F2(x) cos\beta + G2(x) sin\beta] \]
where \(\beta = \frac{x}{\sqrt{2}} + \frac{\pi}{8}\).
Definition at line 195 of file KelvinFunctions.cxx.

static 
Utility function appearing in the asymptotic expansions of DBer(x) and DBei(x).
M(x) is given by
\[ M(x) = \frac{e^{x/\sqrt{2}}}{\sqrt{2\pi x}}\left(1 + \frac{1}{8\sqrt{2} x} + \frac{1}{256 x^{2}}  \frac{399}{6144\sqrt{2} x^{3}} + O\left(\frac{1}{x^{4}}\right)\right) \]
Definition at line 603 of file KelvinFunctions.cxx.

static 
Utility function appearing in the asymptotic expansions of DKer(x) and DKei(x).
N(x) is given by
\[ N(x) = \sqrt{\frac{\pi}{2x}} e^{x/\sqrt{2}} \left(1  \frac{1}{8\sqrt{2} x} + \frac{1}{256 x^{2}} + \frac{399}{6144\sqrt{2} x^{3}} + O\left(\frac{1}{x^{4}}\right)\right) \]
Definition at line 635 of file KelvinFunctions.cxx.

static 
Utility function appearing in the asymptotic expansions of DKer(x) and DKei(x).
\(\phi(x)\) is given by
\[ \phi(x) =  \frac{x}{\sqrt{2}}  \frac{\pi}{8} + \frac{1}{8\sqrt{2} x}  \frac{1}{16 x^{2}} + \frac{25}{384\sqrt{2} x^{3}} + O\left(\frac{1}{x^{5}}\right) \]
Definition at line 651 of file KelvinFunctions.cxx.

static 
Utility function appearing in the asymptotic expansions of DBer(x) and DBei(x).
\(\theta(x)\) is given by
\[ \theta(x) = \frac{x}{\sqrt{2}}  \frac{\pi}{8}  \frac{1}{8\sqrt{2} x}  \frac{1}{16 x^{2}}  \frac{25}{384\sqrt{2} x^{3}} + O\left(\frac{1}{x^{5}}\right) \]
Definition at line 619 of file KelvinFunctions.cxx.

staticprotected 
Definition at line 68 of file KelvinFunctions.h.

staticprotected 
Definition at line 67 of file KelvinFunctions.h.