Kelvin functions of order 0#
Kelvin function \(\mathrm{ber0}(x)\)#
- math53.kelvin_ber0(x)#
Returns the Kelvin function ber. See also Wikipedia [1382], MathWorld [1023], NIST [445], Mpmath [709].
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 \(\mathrm{bei0}(x)\)#
- math53.kelvin_bei0(x)#
Returns the Kelvin function bei. See also Wikipedia [1381], MathWorld [1022], NIST [445], Mpmath [708].
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 \(\mathrm{ker0}(x)\)#
- math53.kelvin_ker0(x)#
Returns the Kelvin function ker. See also Wikipedia [1384], MathWorld [1025], NIST [445], Mpmath [711].
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 \(\mathrm{kei0}(x)\)#
- math53.kelvin_kei0(x)#
Returns the Kelvin function kei. See also Wikipedia [1383], MathWorld [1024], NIST [445], Mpmath [710].
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%)