Kelvin functions

Kelvin function \(\mathrm{ber}(\nu, x)\)

ctx.kelvin_ber(nu, z, scaled=False)

Returns the Kelvin function \(\mathrm{ber}(\nu, x)\).

If scaled is True, then \(\mathrm{ber}(\nu, x) \cdot \exp(-|x| / \sqrt{2})\) is returned.

See also Wikipedia [1499], MathWorld [1098], NIST [467], Mpmath [756].

Here \(\nu\) and \(x\) are, in general, complex numbers. However, if ctx is math53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{R}\) is exspected. If ctx is cmath53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{C}\) is exspected. If ctx is ctxflint then \(n, x \in \mathbb{C}\) is accepted.

This function is traditionally defined for \(\nu \in \mathbb{R}\) and real \(x \ge 0\) as the real part of the Bessel \(J\) function of a rotated argument:

\[\mathrm{ber}(\nu, x) = \Re \Big( J_{\nu}\big( x e^{3\pi i/4} \bigr) \Bigr).\]

In XlCalcNet this is generalized to complex \(\nu\) and \(x\) following the definition given in Maple [387] as

\[\mathrm{ber}(\nu, x) = \frac{ J_{\nu}(x(-a + i a)) + J_{\nu}(x(-a - i a)) }{2}, \quad \text{where } a = \tfrac{1}{2} \sqrt{2},\]

which is equivalent to the traditional definition for \(\nu \in \mathbb{R}\) and real \(x \ge 0\).

Note that this differs from Mpmath, which uses the conventions of Mathematica.

The Kelvin functions are all real valued for real \(x\) and positive \(\nu\).

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%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; z = '3 + 4j'
>>> \mathrm{d}z = dec.kelvinber(n, z); mz = mpm.kelvinber(n, z); gz = gmp.kelvinber(n, z)
>>> fz = fpm.kelvinber(n, z); az = apm.kelvinber(n, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 1.0336162101792974294E+1              + 7.7667689132221259171E+0j
mpm: 1.0336162101792974294e+1              + 7.7667689132221259171e+0j
gmp: 1.0336162101792974294E+01             + 7.7667689132221259171E+00j
fpm: 1.03361621017930E+01                  + 7.76676891322213E+00j
apm: 1.0336162101792974294e+1 (2.032e-18%) + 7.7667689132221259171e+0 (2.225e-18%)j

Kelvin function \(\mathrm{bei}(\nu, x)\)

ctx.kelvin_bei(nu, z, scaled=False)

Returns the Kelvin function \(\mathrm{bei}(\nu, x)\).

If scaled is True, then \(\mathrm{bei}(\nu, x) \cdot \exp(-|x| / \sqrt{2})\) is returned.

See also Wikipedia [1498], MathWorld [1097], NIST [467], Mpmath [755].

Here \(\nu\) and \(x\) are, in general, complex numbers. However, if ctx is math53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{R}\) is exspected. If ctx is cmath53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{C}\) is exspected. If ctx is ctxflint then \(n, x \in \mathbb{C}\) is accepted.

This function is traditionally defined for \(\nu \in \mathbb{R}\) and real \(x \ge 0\) as the imaginary part of the Bessel \(J\) function of a rotated argument:

\[\mathrm{bei}(\nu, x) = \Im \Big( J_{\nu}\big( x e^{3\pi i/4} \bigr) \Bigr).\]

In XlCalcNet this is generalized to complex \(\nu\) and \(x\) following the definition given in Maple [387] as

\[\mathrm{bei}(\nu, x) = \frac{ J_{\nu}(x(-a + i a)) - J_{\nu}(x(-a - i a)) }{2}, \quad \text{where } a = \tfrac{1}{2 i} \sqrt{2},\]

which is equivalent to the traditional definition for \(\nu \in \mathbb{R}\) and real \(x \ge 0\).

Note that this differs from Mpmath, which uses the conventions of Mathematica.

The Kelvin functions are all real valued for real \(x\) and positive \(\nu\).

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%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; z = '3 + 4j'
>>> \mathrm{d}z = dec.kelvinbei(n, z); mz = mpm.kelvinbei(n, z); gz = gmp.kelvinbei(n, z)
>>> fz = fpm.kelvinbei(n, z); az = apm.kelvinbei(n, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: -7.5192613426702233947E+0               + 1.0562898806171521533E+1j
mpm: -7.5192613426702233947e+0               + 1.0562898806171521533e+1j
gmp: -7.5192613426702233947E+00              + 1.0562898806171521533E+01j
fpm: -7.51926134267022E+00                   + 1.05628988061715E+01j
apm: -7.5192613426702233946e+0 (-2.298e-18%) + 1.0562898806171521533e+1 (1.989e-18%)j

Kelvin function \(\mathrm{ker}(\nu, x)\)

ctx.kelvin_ker(nu, z, scaled=False)

Returns the Kelvin function \(\mathrm{ker}(\nu, x)\).

If scaled is True, then \(\mathrm{bei}(\nu, x) \cdot \exp(|x| / \sqrt{2})\) is returned.

See also Wikipedia [1501], MathWorld [1100], NIST [467], Mpmath [758].

Here \(\nu\) and \(x\) are, in general, complex numbers. However, if ctx is math53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{R}\) is exspected. If ctx is cmath53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{C}\) is exspected. If ctx is ctxflint then \(n, x \in \mathbb{C}\) is accepted.

This function is traditionally defined for \(\nu \in \mathbb{R}\) and real \(x \ge 0\) as the real part of the Bessel \(K\) function of a rotated argument:

\[\mathrm{ker}(\nu, x) = \Re \Big( e^{-\pi i/2} K_n\big(x e^{3\pi i/4} \bigr) \Bigr).\]

In XlCalcNet this is generalized to complex \(\nu\) and \(x\) following the definition given in Maple [387] as

\[\mathrm{ker}(\nu, x) = \frac{ e^{-i \nu \pi/2} K_{\nu}(x(a + i a)) + e^{i \nu \pi/2} K_{\nu}(x(a - i a)) }{2}, \quad \text{where } a = \tfrac{1}{2} \sqrt{2},\]

which is equivalent to the traditional definition for \(\nu \in \mathbb{R}\) and real \(x \ge 0\).

Note that this differs from Mpmath, which uses the conventions of Mathematica.

The Kelvin functions are all real valued for real \(x\) and positive \(\nu\).

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%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; z = '3 + 4j'
>>> \mathrm{d}z = dec.kelvinker(n, z); mz = mpm.kelvinker(n, z); gz = gmp.kelvinker(n, z)
>>> fz = fpm.kelvinker(n, z); az = apm.kelvinker(n, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 5.0016351579287135729E-1              + 2.7244626208949455251E-1j
mpm: 5.0016351579287135729e-1              + 2.7244626208949455251e-1j
gmp: 5.0016351579287135729E-01             + 2.7244626208949455251E-01j
fpm: 5.00163515792871E-01                  + 2.72446262089495E-01j
apm: 5.0016351579287135710e-1 (2.917e-15%) + 2.7244626208949455183e-1 (4.106e-15%)j

Kelvin function \(\mathrm{kei}(\nu, x)\)

ctx.kelvin_kei(nu, z, scaled=False)

Returns the Kelvin function \(\mathrm{kei}(\nu, x)\).

If scaled is True, then \(\mathrm{kei}(\nu, x) \cdot \exp(|x| / \sqrt{2})\) is returned.

See also Wikipedia [1500], MathWorld [1099], NIST [467], Mpmath [757].

Here \(\nu\) and \(x\) are, in general, complex numbers. However, if ctx is math53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{R}\) is exspected. If ctx is cmath53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{C}\) is exspected. If ctx is ctxflint then \(n, x \in \mathbb{C}\) is accepted.

This function is traditionally defined for \(\nu \in \mathbb{R}\) and real \(x \ge 0\) as the imaginary part of the Bessel \(K\) function of a rotated argument:

\[\mathrm{kei}(\nu, x) = \Im \Big( e^{-\pi i/2} K_n\big(x e^{3\pi i/4} \bigr) \Bigr).\]

In XlCalcNet this is generalized to complex \(\nu\) and \(x\) following the definition given in Maple [387] as

\[\mathrm{kei}(\nu, x) = \frac{ e^{-i \nu \pi/2} K_{\nu}(x(a + i a)) - e^{i \nu \pi/2} K_{\nu}(x(a - i a)) }{2 i}, \quad \text{where } a = \tfrac{1}{2} \sqrt{2}.\]

which is equivalent to the traditional definition for \(\nu \in \mathbb{R}\) and real \(x \ge 0\).

Note that this differs from Mpmath, which uses the conventions of Mathematica.

The Kelvin functions are all real valued for real \(x\) and positive \(\nu\).

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%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; z = '3 + 4j'
>>> \mathrm{d}z = dec.kelvinkei(n, z); mz = mpm.kelvinkei(n, z); gz = gmp.kelvinkei(n, z)
>>> fz = fpm.kelvinkei(n, z); az = apm.kelvinkei(n, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 2.7516275915865256214E-1              - 4.9739028524739862760E-1j
mpm: 2.7516275915865256214e-1              - 4.9739028524739862760e-1j
gmp: 2.7516275915865256214E-01             - 4.9739028524739862760E-01j
fpm: 2.75162759158653E-01                  - 4.97390285247399E-01j
apm: 2.7516275915865859000e-1 (2.965e-11%) - 4.9739028524740313000e-1 (-2.154e-11%)j

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

ctx.kelvin_ber_prime(nu, z, scaled=False)

Returns \(\mathrm{ber}'(\nu, x)\), the first derivative (with respect to \(x\)) of the Kelvin function \(\mathrm{ber}(\nu, x)\).

If scaled is True, then \(\mathrm{ber}(\nu, x) \cdot \exp(-|x| / \sqrt{2})\) is returned.

See also Wikipedia [1499], MathWorld [1098], NIST [467], Mpmath [756].

Here \(\nu\) and \(x\) are, in general, complex numbers. However, if ctx is math53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{R}\) is exspected. If ctx is cmath53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{C}\) is exspected. If ctx is ctxflint then \(n, x \in \mathbb{C}\) is accepted.

The function is calculated as

\[\mathrm{ber}'(\nu, x) = \frac{a_1 J'_{\nu}(x \cdot a_1) + a_2 J'_{\nu}(x \cdot a_2) }{2},\]

where \(a = \tfrac{1}{2} \sqrt{2}\), \(a_1 = -a + i a\), \(a_2 = -a - i a\), and \(J'_{\nu}(x)\) is the first derivative (with respect to \(x\)) of the Bessel function \(J_{\nu}(x)\).

The Kelvin functions are all real valued for real \(x\) and positive \(\nu\).

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%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; z = '3 + 4j'
>>> \mathrm{d}z = dec.kelvinber(n, z); mz = mpm.kelvinber(n, z); gz = gmp.kelvinber(n, z)
>>> fz = fpm.kelvinber(n, z); az = apm.kelvinber(n, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 1.0336162101792974294E+1              + 7.7667689132221259171E+0j
mpm: 1.0336162101792974294e+1              + 7.7667689132221259171e+0j
gmp: 1.0336162101792974294E+01             + 7.7667689132221259171E+00j
fpm: 1.03361621017930E+01                  + 7.76676891322213E+00j
apm: 1.0336162101792974294e+1 (2.032e-18%) + 7.7667689132221259171e+0 (2.225e-18%)j

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

ctx.kelvin_bei_prime(nu, z, scaled=False)

Returns \(\mathrm{bei}'(\nu, x)\), the first derivative (with respect to \(x\)) of the Kelvin function \(\mathrm{bei}(\nu, x)\).

If scaled is True, then \(\mathrm{bei}(\nu, x) \cdot \exp(-|x| / \sqrt{2})\) is returned.

See also Wikipedia [1499], MathWorld [1098], NIST [467], Mpmath [756].

Here \(\nu\) and \(x\) are, in general, complex numbers. However, if ctx is math53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{R}\) is exspected. If ctx is cmath53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{C}\) is exspected. If ctx is ctxflint then \(n, x \in \mathbb{C}\) is accepted.

The function is calculated as

\[\mathrm{bei}'(\nu, x) = \frac{a_1 J'_{\nu}(x \cdot a_1) - a_2 J'_{\nu}(x \cdot a_2) }{2 i},\]

where \(a = \tfrac{1}{2} \sqrt{2}\), \(a_1 = -a + i a\), \(a_2 = -a - i a\), and \(J'_{\nu}(x)\) is the first derivative (with respect to \(x\)) of the Bessel function \(J_{\nu}(x)\).

The Kelvin functions are all real valued for real \(x\) and positive \(\nu\).

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%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; z = '3 + 4j'
>>> \mathrm{d}z = dec.kelvinbei(n, z); mz = mpm.kelvinbei(n, z); gz = gmp.kelvinbei(n, z)
>>> fz = fpm.kelvinbei(n, z); az = apm.kelvinbei(n, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: -7.5192613426702233947E+0               + 1.0562898806171521533E+1j
mpm: -7.5192613426702233947e+0               + 1.0562898806171521533e+1j
gmp: -7.5192613426702233947E+00              + 1.0562898806171521533E+01j
fpm: -7.51926134267022E+00                   + 1.05628988061715E+01j
apm: -7.5192613426702233946e+0 (-2.298e-18%) + 1.0562898806171521533e+1 (1.989e-18%)j

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

ctx.kelvin_ker_prime(nu, z, scaled=False)

Returns \(\mathrm{ker}'(\nu, x)\), the first derivative (with respect to \(x\)) of the Kelvin function \(\mathrm{ker}(\nu, x)\).

If scaled is True, then \(\mathrm{ber}(\nu, x) \cdot \exp(|x| / \sqrt{2})\) is returned.

See also Wikipedia [1501], MathWorld [1100], NIST [467], Mpmath [758].

Here \(\nu\) and \(x\) are, in general, complex numbers. However, if ctx is math53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{R}\) is exspected. If ctx is cmath53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{C}\) is exspected. If ctx is ctxflint then \(n, x \in \mathbb{C}\) is accepted.

The function is calculated as

\[\mathrm{ker}'(\nu, x) = \frac{e^{-\nu\pi i/2} a_1 K'_{\nu}(x \cdot a_1) + e^{\nu\pi i/2} a_2 K'_{\nu}(x \cdot a_2) }{2},\]

where \(a = \tfrac{1}{2} \sqrt{2}, \: a_1 = -a + i a, \:a_2 = -a - i a\), and \(K'_{\nu}(x)\) is the first derivative (with respect to \(x\)) of the Bessel function \(K_{\nu}(x)\).

The Kelvin functions are all real valued for real \(x\) and positive \(\nu\).

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%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; z = '3 + 4j'
>>> \mathrm{d}z = dec.kelvinker(n, z); mz = mpm.kelvinker(n, z); gz = gmp.kelvinker(n, z)
>>> fz = fpm.kelvinker(n, z); az = apm.kelvinker(n, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 5.0016351579287135729E-1              + 2.7244626208949455251E-1j
mpm: 5.0016351579287135729e-1              + 2.7244626208949455251e-1j
gmp: 5.0016351579287135729E-01             + 2.7244626208949455251E-01j
fpm: 5.00163515792871E-01                  + 2.72446262089495E-01j
apm: 5.0016351579287135710e-1 (2.917e-15%) + 2.7244626208949455183e-1 (4.106e-15%)j

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

ctx.kelvin_kei_prime(nu, z, scaled=False)

Returns \(\mathrm{kei}'(\nu, x)\), the first derivative (with respect to \(x\)) of the Kelvin function \(\mathrm{kei}(\nu, x)\).

If scaled is True, then \(\mathrm{kei}(\nu, x) \cdot \exp(|x| / \sqrt{2})\) is returned.

See also Wikipedia [1500], MathWorld [1099], NIST [467], Mpmath [757].

Here \(\nu\) and \(x\) are, in general, complex numbers. However, if ctx is math53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{R}\) is exspected. If ctx is cmath53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{C}\) is exspected. If ctx is ctxflint then \(n, x \in \mathbb{C}\) is accepted.

The function is calculated as

\[\mathrm{ker}'(\nu, x) = \frac{e^{-\nu\pi i/2} a_1 K'_{\nu}(x \cdot a_1) - e^{\nu\pi i/2} a_2 K'_{\nu}(x \cdot a_2) }{2 i},\]

where \(a = \tfrac{1}{2} \sqrt{2}, \: a_1 = -a + i a, \:a_2 = -a - i a\), and \(K'_{\nu}(x)\) is the first derivative (with respect to \(x\)) of the Bessel function \(K_{\nu}(x)\).

The Kelvin functions are all real valued for real \(x\) and positive \(\nu\).

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%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; z = '3 + 4j'
>>> \mathrm{d}z = dec.kelvinkei(n, z); mz = mpm.kelvinkei(n, z); gz = gmp.kelvinkei(n, z)
>>> fz = fpm.kelvinkei(n, z); az = apm.kelvinkei(n, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 2.7516275915865256214E-1              - 4.9739028524739862760E-1j
mpm: 2.7516275915865256214e-1              - 4.9739028524739862760e-1j
gmp: 2.7516275915865256214E-01             - 4.9739028524739862760E-01j
fpm: 2.75162759158653E-01                  - 4.97390285247399E-01j
apm: 2.7516275915865859000e-1 (2.965e-11%) - 4.9739028524740313000e-1 (-2.154e-11%)j