Bessel functions of integer order#
Bessel function of the 1st kind, order \(0, J_{0}(x)\)#
- ctx.bessel_j0(x)#
where
ctxismath53orctxcpp.Returns \(J_0(x)\), the Bessel function of the 1st kind, order zero. See also Wikipedia [1356], MathWorld [951], NIST [463], BoostMath [100], Ehrhardt [309] (3.1.1.1).
An example in Python
>>> from xlcalcnet import xreal >>> xreal.BesselJ0(0.5) xreal('5.2359877559829887307E-1') >>> xreal.BesselJ0('0.51') xreal('5.3518479027559984754E-1')
An example in Visual Basic
>>> from xlcalcnet import Gpr >>> Gpr.BesselJ0(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.BesselJ0('0.51') Gpr('5.3518479027559984754E-1')
Bessel function of the 1st kind, order \(1, J_{1}(x)\)#
- ctx.bessel_j1(x)#
where
ctxismath53orctxcpp.Returns \(J_1(x)\), the Bessel function of the 1st kind, order one. See also Wikipedia [1356], MathWorld [951], NIST [463], BoostMath [100], Ehrhardt [309] (3.1.1.1).
An example in Python
>>> from xlcalcnet import xreal >>> xreal.BesselJ1(0.5) xreal('5.2359877559829887307E-1') >>> xreal.BesselJ1('0.51') xreal('5.3518479027559984754E-1')
An example in Visual Basic
>>> from xlcalcnet import Gpr >>> Gpr.BesselJ1(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.BesselJ1('0.51') Gpr('5.3518479027559984754E-1')
Bessel function of the 1st kind, integer order \(n, J_{n}(x)\)#
- ctx.bessel_jn(n, x)#
where
ctxismath53orctxcpp.Returns \(J_n(x)\), the Bessel function of the 1st kind, order \(n\). See also Wikipedia [1356], MathWorld [951], NIST [463], BoostMath [100], Ehrhardt [309] (3.1.1.1).
\[J_{n}(x) = \left(\tfrac{1}{2}x\right)^{n} \sum_{k=0}^\infty (-1)^k \frac{(x^2 / 4)^k}{k! \Gamma(n+k+1)}.\]An example in Python
>>> from xlcalcnet import xreal >>> xreal.BesselJn(3, 0.5) xreal('5.2359877559829887307E-1') >>> xreal.BesselJn(3, '0.51') xreal('5.3518479027559984754E-1')
An example in Visual Basic
>>> from xlcalcnet import Gpr >>> Gpr.BesselJn(3, 0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.BesselJn(3, '0.51') Gpr('5.3518479027559984754E-1')
Bessel function of the 2nd kind, order \(0, Y_{0}(x)\)#
- ctx.bessel_y0(x)#
where
ctxismath53orctxcpp.Returns \(Y_0(x)\), the Bessel function of the 2nd kind, order zero, \(x>0\). See also Wikipedia [1357], MathWorld [952], NIST [464], BoostMath [100], Ehrhardt [309] (3.1.1.1).
An example in Python
>>> from xlcalcnet import xreal >>> xreal.BesselY0(0.5) xreal('5.2359877559829887307E-1') >>> xreal.BesselY0('0.51') xreal('5.3518479027559984754E-1')
An example in Visual Basic
>>> from xlcalcnet import Gpr >>> Gpr.BesselY0(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.BesselY0('0.51') Gpr('5.3518479027559984754E-1')
Bessel function of the 2nd kind, order \(1, Y_{1}(x)\)#
- ctx.bessel_y1(x)#
where
ctxismath53orctxcpp.Returns \(Y_1(x)\), the Bessel function of the 2nd kind, order one, \(x>0\). See also Wikipedia [1357], MathWorld [952], NIST [464], BoostMath [100], Ehrhardt [309] (3.1.1.1).
An example in Python
>>> from xlcalcnet import xreal >>> xreal.BesselY1(0.5) xreal('5.2359877559829887307E-1') >>> xreal.BesselY1('0.51') xreal('5.3518479027559984754E-1')
An example in Visual Basic
>>> from xlcalcnet import Gpr >>> Gpr.BesselY1(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.BesselY1('0.51') Gpr('5.3518479027559984754E-1')
Bessel function of the 2nd kind, integer order \(n, Y_{n}(x)\)#
- ctx.bessel_yn(x)#
where
ctxismath53orctxcpp.Returns \(Y_n(x)\), the Bessel function of the 2nd kind, order n, \(x>0\). See also Wikipedia [1357], MathWorld [952], NIST [464], BoostMath [100], Ehrhardt [309] (3.1.1.1).
\[Y_{n}(z)=-{\frac {\left({\frac {z}{2}}\right)^{-n}}{\pi }}\sum _{k=0}^{n-1}{\frac {(n-k-1)!}{k!}}\left({\frac {z^{2}}{4}}\right)^{k}+{\frac {2}{\pi }}J_{n}(z)\log {\frac {z}{2}}-{\frac {\left({\frac {z}{2}}\right)^{n}}{\pi }}\sum _{k=0}^{\infty }(\psi (k+1)+\psi (n+k+1)){\frac {\left(-{\frac {z^{2}}{4}}\right)^{k}}{k!(n+k)!}}\]An example in Python
>>> from xlcalcnet import xreal >>> xreal.BesselYn(3, 0.5) xreal('5.2359877559829887307E-1') >>> xreal.BesselYn(3, '0.51') xreal('5.3518479027559984754E-1')
An example in double precision (32 bit version of xlcalcnet)
>>> from xlcalcnet import Gpr >>> Gpr.BesselYn(3, 0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.BesselYn(3, '0.51') Gpr('5.3518479027559984754E-1')
Bessel Lambda function, \(\Lambda(\nu,x)\)#
- math53.bessel_lambda(v, x)#
Returns the Bessel \(\Lambda_{\nu}(x)\) function, defined for \(x, \nu \ge 0\) as
\[\Lambda(\nu,x) = \Gamma(\nu+1) \frac{J_{\nu}(x)}{(x/2)^{\nu}}\]See also: Ehrhardt [309] (3.1.3.3).
An example in Python
>>> from xlcalcnet import xreal >>> xreal.BesselLambda(3, 0.5) xreal('5.2359877559829887307E-1') >>> xreal.BesselLambda(3, '0.51') xreal('5.3518479027559984754E-1')
An example in Visual Basic
>>> from xlcalcnet import Gpr >>> Gpr.BesselLambda(3, 0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.BesselLambda(3, '0.51') Gpr('5.3518479027559984754E-1')