Modified Bessel functions of integer order#

Modified Bessel function of the 1st kind, order \(0, I_{0}(x)\)#

math53.bessel_i0(x)#

Returns I0(x), the modified Bessel function of the 1st kind, order zero. See also Wikipedia [1370], MathWorld [980], NIST [468], BoostMath [114], Ehrhardt [309] (3.1.2.1).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.BesselI0(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.BesselI0('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.BesselI0(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.BesselI0('0.51')
Gpr('5.3518479027559984754E-1')

Modified Bessel function of the 1st kind, order \(1, I_{1}(x)\)#

math53.bessel_i1(x)#

Returns I1(x), the modified Bessel function of the 1st kind, order one. See also Wikipedia [1370], MathWorld [980], NIST [468], BoostMath [114], Ehrhardt [309] (3.1.2.3).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.BesselI1(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.BesselI1('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.BesselI1(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.BesselI1('0.51')
Gpr('5.3518479027559984754E-1')

Modified Bessel function of the 1st kind, integer order \(n, I_{n}(x)\)#

math53.damath_bessel_in(x)#

Returns I_n(x), the modified Bessel function of the 1st kind, order n. See also Wikipedia [1370], MathWorld [980], NIST [468], BoostMath [114], Ehrhardt [309] (3.1.2.5).

\[I_{\nu}\left(z\right)=(\tfrac{1}{2}z)^{\nu}\sum_{k=0}^{\infty}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!\Gamma\left(\nu+k+1\right)}.\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.BesselIn(3, 0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.BesselIn(3, '0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.BesselIn(3, 0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.BesselIn(3, '0.51')
Gpr('5.3518479027559984754E-1')

Modified Bessel function of the 2nd kind, order 0, \(K_{0}(x)\)#

math53.bessel_k0(x)#

Returns K0(x), the modified Bessel function of the 2nd kind, order zero, x>0. See also Wikipedia [1370], MathWorld [981], NIST [469], BoostMath [114], Ehrhardt [309] (3.1.2.6).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.BesselK0(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.BesselK0('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.BesselK0(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.BesselK0('0.51')
Gpr('5.3518479027559984754E-1')

Modified Bessel function of the 2nd kind, order \(1, K_{1}(x)\)#

math53.bessel_k1(x)#

Returns K1(x), the modified Bessel function of the 2nd kind, order one, x>0. See also Wikipedia [1370], MathWorld [981], NIST [469], BoostMath [114], Ehrhardt [309] (3.1.2.8).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.BesselK1(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.BesselK1('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.BesselK1(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.BesselK1('0.51')
Gpr('5.3518479027559984754E-1')

Modified Bessel function of the 2nd kind, integer order \(n, K_{n}(x)\)#

math53.bessel_kn(x)#

Returns K_n(x), the modified Bessel function of the 2nd kind, order n, x > 0. See also Wikipedia [1370], MathWorld [981], NIST [469], BoostMath [114], Ehrhardt [309] (3.1.2.10).

\begin{eqnarray} K_{n}\left(z\right) & = & \tfrac{1}{2}(\tfrac{1}{2}z)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}(-\tfrac{1}{4}z^{2})^{k}+(-1)^{n+1}\log\left(\tfrac{1}{2}z\right)I_{n}\left(z\right) \\ & + & (-1)^{n}\tfrac{1}{2}(\tfrac{1}{2}z)^{n}\sum_{k=0}^{\infty}\left(\psi\left(k+1\right)+\psi\left(n+k+1\right)\right)\frac{(\tfrac{1}{4}z^{2})^{k}}{k!(n+k)!} \nonumber \end{eqnarray}

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.BesselKn(3, 0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.BesselKn(3, '0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.BesselKn(3, 0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.BesselKn(3, '0.51')
Gpr('5.3518479027559984754E-1')

Exponentially scaled modified Bessel function of the 1st kind, order \(0, I_{0,e}(x)\)#

math53.bessel_i0e(x)#

Returns I0(x)*exp(-\(|x|\)), the exponentially scaled modified Bessel function of the 1st kind, order zero. See also Wikipedia [1370], MathWorld [980], NIST [468], BoostMath [114], Ehrhardt [309] (3.1.2.2).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.BesselI0e(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.BesselI0e('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.BesselI0e(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.BesselI0e('0.51')
Gpr('5.3518479027559984754E-1')

Exponentially scaled modified Bessel function of the 1st kind, order \(1, I_{1,e}(x)\)#

math53.bessel_i1e(x)#

Returns I1(x)*exp(-\(|x|\)), the exponentially scaled modified Bessel function of the 1st kind, order one. See also Wikipedia [1370], MathWorld [980], NIST [468], BoostMath [114], Ehrhardt [309] (3.1.2.4).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.BesselI1e(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.BesselI1e('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.BesselI1e(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.BesselI1e('0.51')
Gpr('5.3518479027559984754E-1')

Exponentially scaled modified Bessel function of the 2nd kind, order 0, \(K_{0,e}(x)\)#

math53.bessel_k0e(x)#

Returns K0(x)*exp(x), the exponentially scaled modified Bessel function of the 2nd kind, order zero, x>0. See also Wikipedia [1370], MathWorld [981], NIST [469], BoostMath [114], Ehrhardt [309] (3.1.2.7).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.BesselK0e(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.BesselK0e('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.BesselK0e(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.BesselK0e('0.51')
Gpr('5.3518479027559984754E-1')

Exponentially scaled modified Bessel function of the 2nd kind, order \(1, K_{1,e}(x)\)#

math53.bessel_k1e(x)#

Returns K1(x)*exp(x), the exponentially scaled modified Bessel function of the 2nd kind, order one, x>0. See also Wikipedia [1370], MathWorld [981], NIST [469], BoostMath [114], Ehrhardt [309] (3.1.2.9).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.BesselK1e(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.BesselK1e('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.BesselK1e(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.BesselK1e('0.51')
Gpr('5.3518479027559984754E-1')

Modified Bessel functions of integer order

Contents

Modified Bessel functions of integer order#

Modified Bessel function of the 1st kind, order \(0, I_{0}(x)\)#

Modified Bessel function of the 1st kind, order \(1, I_{1}(x)\)#

Modified Bessel function of the 1st kind, integer order \(n, I_{n}(x)\)#

Modified Bessel function of the 2nd kind, order 0, \(K_{0}(x)\)#

Modified Bessel function of the 2nd kind, order \(1, K_{1}(x)\)#

Modified Bessel function of the 2nd kind, integer order \(n, K_{n}(x)\)#

Exponentially scaled modified Bessel function of the 1st kind, order \(0, I_{0,e}(x)\)#

Exponentially scaled modified Bessel function of the 1st kind, order \(1, I_{1,e}(x)\)#

Exponentially scaled modified Bessel function of the 2nd kind, order 0, \(K_{0,e}(x)\)#

Exponentially scaled modified Bessel function of the 2nd kind, order \(1, K_{1,e}(x)\)#