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class ROOT::Math::KelvinFunctions
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library: libMathMore
#include "KelvinFunctions.h"
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class ROOT::Math::KelvinFunctions








Function Members (Methods)


public:



 virtual~KelvinFunctions()


static doubleBei(double x)


static doubleBer(double x)


static doubleDBei(double x)


static doubleDBer(double x)


static doubleDKei(double x)


static doubleDKer(double x)


static doubleF1(double x)


static doubleF2(double x)


static doubleG1(double x)


static doubleG2(double x)


static doubleKei(double x)


ROOT::Math::KelvinFunctionsKelvinFunctions()


ROOT::Math::KelvinFunctionsKelvinFunctions(const ROOT::Math::KelvinFunctions&)


static doubleKer(double x)


static doubleM(double x)


static doubleN(double x)


ROOT::Math::KelvinFunctions&operator=(const ROOT::Math::KelvinFunctions&)


static doublePhi(double x)


static doubleTheta(double x)










Data Members


protected:



static doublefgEpsilon


static doublefgMin







Class Charts






Inheritance
Inherited Members
Includes
Libraries



Class Charts


Function documentation


double  Ber(double x)


Ber(x) = Ber_{0}(x) = Re#left[J_{0}#left(x e^{3#pii/4}#right)#right]
 where x is real, and J_{0}(z) is the zeroth-order 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#pix}} [F1(x) cos#alpha + G1(x) sin#alpha] - #frac{1}{#pi}Kei(x)
 where #alpha = #frac{x}{#sqrt{2}} - #frac{#pi}{8}.
 See also F1(x) and G1(x).


output of MACRO_ROOT::Math::KelvinFunctions_Ber_1_5_c





double  Bei(double x)


Bei(x) = Bei_{0}(x) = Im#left[J_{0}#left(x e^{3#pii/4}#right)#right]
 where x is real, and J_{0}(z) is the zeroth-order 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#pix}} [F1(x) sin#alpha + G1(x) cos#alpha] - #frac{1}{#pi}Ker(x)
 where #alpha = #frac{x}{#sqrt{2}} - #frac{#pi}{8}
 See also F1(x) and G1(x).


output of MACRO_ROOT::Math::KelvinFunctions_Bei_1_5_c





double  Ker(double x)


Ker(x) = Ker_{0}(x) = Re#left[K_{0}#left(x e^{3#pii/4}#right)#right]
 where x is real, and K_{0}(z) is the zeroth-order 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 Euler-Mascheroni 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}
 See also F2(x) and G2(x).


output of MACRO_ROOT::Math::KelvinFunctions_Ker_1_8_c





double  Kei(double x)


Kei(x) = Kei_{0}(x) = Im#left[K_{0}#left(x e^{3#pii/4}#right)#right]
 where x is real, and K_{0}(z) is the zeroth-order 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 Euler-Mascheroni 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}
 See also F2(x) and G2(x).


output of MACRO_ROOT::Math::KelvinFunctions_Kei_1_8_c





double  DBer(double x)


 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)
 See also M(x) and Theta(x).


output of MACRO_ROOT::Math::KelvinFunctions_DBer_1_1_c





double  DBei(double x)


 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)
 See also M(x) and Theta(x).


output of MACRO_ROOT::Math::KelvinFunctions_DBei_1_1_c





double  DKer(double x)


 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)
 See also N(x) and Phi(x).


output of MACRO_ROOT::Math::KelvinFunctions_DKer_1_1_c





double  DKei(double x)


 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)
 See also N(x) and Phi(x).


output of MACRO_ROOT::Math::KelvinFunctions_DKei_1_1_c





double  F1(double x)


 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)





double  F2(double x)


 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)





double  G1(double x)


 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)





double  G2(double x)


 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)





double  M(double x)


 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#pix}}#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)





double  Theta(double x)


 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)





double  N(double x)


 Utility function appearing in the asymptotic expansions of DKer(x) and
 DKei(x). (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)





double  Phi(double x)


 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)





virtual ~KelvinFunctions()


 Include and empty virtual desctructor to eliminate compiler warnings




{}










Last update: Mon Jun 25 19:39:39 2007





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