Kelvin functions of order 0

Kelvin functions of order 0#

Kelvin function \(\mathrm{ber0}(x)\)#

ctx.kelvin_ber0(x)#

Returns the Kelvin function ber. See also Wikipedia [1499], MathWorld [1098], NIST [467], Mpmath [756].

Returns the Kelvin function \(\mathrm{ber}_0(x)\), defined as

\[\mathrm{ber}_0(x) = \sum_{k=0}^\infty (-1)^k \frac{\left(\tfrac{1}{4}x^2\right)^{2k}}{((2k)!)^2}.\]

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; n = 0; x = 3
>>> \mathrm{d}x = dec.kelvinber(n, x); mx = mpm.kelvinber(n, x); gx = gmp.kelvinber(n, x)
>>> fx = fpm.kelvinber(n, x); ax = apm.kelvinber(n, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax],  aligned=True)
dec: -2.213802495986938888682464345899509922321E-1
mpm: -2.213802495986938888682464345899509922321e-1
gmp: -2.213802495986938888682464345899509922321E-01
fpm: -2.21380249598694E-01
apm: -2.213802495986938888682464345899509922332e-1 (-1.283e-37%)

Kelvin function \(\mathrm{bei0}(x)\)#

ctx.kelvin_bei0(x)#

Returns the Kelvin function bei. See also Wikipedia [1498], MathWorld [1097], NIST [467], Mpmath [755].

Returns the Kelvin function \(\mathrm{bei}_0(x)\) is defined as

\[\mathrm{bei}_0(x) = \sum_{k=0}^\infty (-1)^k \frac{\left(\tfrac{1}{4}x^2\right)^{2k+1}}{((2k+1)!)^2}.\]

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; n = 0; x = 3
>>> \mathrm{d}x = dec.kelvinbei(n, x); mx = mpm.kelvinbei(n, x); gx = gmp.kelvinbei(n, x)
>>> fx = fpm.kelvinbei(n, x); ax = apm.kelvinbei(n, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax],  aligned=True)
dec: 1.937586785266042766896808122272260201255E+0
mpm: 1.937586785266042766896808122272260201255e+0
gmp: 1.937586785266042766896808122272260201255E+00
fpm: 1.93758678526604E+00
apm: 1.937586785266042766896808122272260201256e+0 (1.54e-38%)

Kelvin function \(\mathrm{ker0}(x)\)#

ctx.kelvin_ker0(x)#

Returns the Kelvin function ker. See also Wikipedia [1501], MathWorld [1100], NIST [467], Mpmath [758].

The Kelvin function \(\mathrm{ker}_0(x)\) is defined as

\[\mathrm{ker}_0(x) = -\log \left(\tfrac{1}{2}x \right) \mathrm{ber}_0(x) +\tfrac{1}{4}\pi \; \mathrm{bei}_0(x) + \sum_{k=0}^\infty (-1)^k \frac{\psi(2k+1) \left(\tfrac{1}{4}x^2\right)^{2k}}{((2k)!)^2}.\]

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; n = 0; x = 3
>>> \mathrm{d}x = dec.kelvinker(n, x); mx = mpm.kelvinker(n, x); gx = gmp.kelvinker(n, x)
>>> fx = fpm.kelvinker(n, x); ax = apm.kelvinker(n, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax],  aligned=True)
dec: -6.702923330379869775199782194748322134382E-2
mpm: -6.702923330379869775199782194748322134382e-2
gmp: -6.702923330379869775199782194748322134382E-02
fpm: -6.70292333037987E-02
apm: -6.702923330379869775199782194748322134370e-2 (-8.669e-36%)

Kelvin function \(\mathrm{kei0}(x)\)#

ctx.kelvin_kei0(x)#

Returns the Kelvin function kei. See also Wikipedia [1500], MathWorld [1099], NIST [467], Mpmath [757].

The Kelvin function \(\mathrm{kei}_0(x)\) is defined as

\[\mathrm{kei}_0(x) = -\log \left(\tfrac{1}{2}x \right) \mathrm{ber}_0(x) -\tfrac{1}{4}\pi \; \mathrm{bei}_0(x) + \sum_{k=0}^\infty (-1)^k \frac{\psi(2k+2) \left(\tfrac{1}{4}x^2\right)^{2k+1}}{((2k+1)!)^2}\]

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; n = 0; x = 3
>>> \mathrm{d}x = dec.kelvinkei(n, x); mx = mpm.kelvinkei(n, x); gx = gmp.kelvinkei(n, x)
>>> fx = fpm.kelvinkei(n, x); ax = apm.kelvinkei(n, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax],  aligned=True)
dec: -5.112188404598678140246687753930501705762E-2
mpm: -5.112188404598678140246687753930501705762e-2
gmp: -5.112188404598678140246687753930501705762E-02
fpm: -5.11218840459868E-02
apm: -5.112188404598678140246687753930501705753e-2 (-9.353e-36%)

First derivative of the Kelvin function \(\mathrm{ber0}, \mathrm{ber0}'(x)\)#

ctx.kelvin_ber0_prime(x)#

Returns the Kelvin function ber’(x), x >= 0

See also: Ehrhardt [309] (3.1.8.8).

An example in Python

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

First derivative of the Kelvin function \(\mathrm{bei0}, \mathrm{bei0}'(x)\)#

ctx.kelvin_bei0_prime(x)#

Returns the Kelvin function bei’(x), x >= 0

See also: Ehrhardt [309] (3.1.8.9).

An example in Python

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

An example in Visual Basic

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

First derivative of the Kelvin function \(\mathrm{ker0}, \mathrm{ker0}'(\nu, x)\)#

ctx.kelvin_ker0_prime(x)#

Returns the Kelvin function ker’(x), x > 0

See also: Ehrhardt [309] (3.1.8.10).

An example in Python

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

First derivative of the Kelvin function \(\mathrm{kei0}, \mathrm{kei0}'(x)\)#

ctx.kelvin_kei0_prime(x)#

Returns the Kelvin function kei’(x), x >= 0

See also: Ehrhardt [309] (3.1.8.11).

An example in Python

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

Derivatives of Kelvin functions#

math53.kelvin0Prime(x)#

Returns the derivatives of the Kelvin functions berp = ber0(x), beip = bei0(x), kerp = ker0(x), and keip = kei0(x) for \(x \ge 0\).

See also: Ehrhardt [309] (3.1.8.7).

See NIST for general formulas

An example in Python

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