Hankel functions

Hankel function of the first kind \(H^{(1)}_{\nu}(x)\)

ctx.hankel_h1(nu, x, scaled=False)

where ctx is math53 or ctxboost.

Returns the Hankel function of the first kind, defined as \(\displaystyle H^{(1)}_{\nu}(x) = J_{\nu}(x) + i Y_{\nu}(x)\).

If scaled is True, then \(\displaystyle H^{(1)e}_{\nu}(x) = H^{(1)}_{\nu}(x) \cdot \exp(-i x)\) is returned, except for a real ctx where just \(\displaystyle H^{(1)}_{\nu}(x)\) is returned.

If ctx is math53, ctxboost or ctxflintreal, then \(\nu, x \in \mathbb{R}\) is exspected. If ctx is cmath53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{C}\) is exspected. If ctx is ctxflintcplx then \(\nu, x \in \mathbb{C}\) is accepted.

See also Wikipedia [1412], MathWorld [1013], NIST [466], BoostMath [121], Mpmath [660].

>>> from xlcalcnet import XComplex
>>> XComplex.HankelH1(10.5, 6.3)
XComplex('5.2359877559829887307E-1')
>>> XComplex.HankelH1(10.5, 6.3)
XComplex('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpc
>>> Gpc.HankelH1(10.5, 6.3)
Gpc('5.2359877559829887307E-1')
>>> Gpc.HankelH1(10.5, 6.3)
Gpc('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; n= 10; x = 30
>>> \mathrm{d}x = dec.hankel1(n, x); mx = mpm.hankel1(n, x); gx = gmp.hankel1(n, x)
>>> fx = fpm.hankel1(n, x); ax = apm.hankel1(n, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax],  aligned=True)
dec: -1.2987689399858876819E-1               + 7.5056702122397113289E-2j
mpm: -1.2987689399858876819e-1               + 7.5056702122397113289e-2j
gmp: -1.2987689399858876819E-01              + 7.5056702122397113289E-02j
fpm: -1.29876893998589E-01                   + 7.50567021223971E-02j
apm: -1.2987689399889094748e-1 (-9.932e-10%) + 7.5056702113900822249e-2 (2.838e-6%)j

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; n= 10; z = '3 + 4j'
>>> \mathrm{d}z = dec.hankel1(n, z); mz = mpm.hankel1(n, z); gz = gmp.hankel1(n, z)
>>> fz = fpm.hankel1(n, z); az = apm.hankel1(n, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: -6.9984001073685610955E+0              + 6.7915518863025118064E+0j
mpm: -6.9984001073685610955e+0              + 6.7915518863025118064e+0j
gmp: -6.9984001073685610955E+00             + 6.7915518863025118064E+00j
fpm: -6.99840010736856E+00                  + 6.79155188630251E+00j
apm: -6.9984001073685610955e+0 (-2.13e-18%) + 6.7915518863025118064e+0 (2.145e-18%)j

Hankel function of the second kind \(H^{(2)}_{\nu}(x)\)

ctx.hankel_h2(nu, x, scaled=False)

where ctx is math53 or ctxboost.

Returns the Hankel function of the second kind, defined as \(\displaystyle H^{(2)}_{\nu}(x) = J_{\nu}(x) - i Y_{\nu}(x)\).

If scaled is True, then \(\displaystyle H^{(2)e}_{\nu}(x) = H^{(2)}_{\nu}(x) \cdot \exp(i x)\) is returned, except for a real ctx where just \(\displaystyle H^{(2)}_{\nu}(x)\) is returned.

If ctx is math53, ctxboost or ctxflintreal, then \(\nu, x \in \mathbb{R}\) is exspected. If ctx is cmath53, then \(\nu \in \mathbb{R}\) and \(x \in \mathbb{C}\) is exspected. If ctx is ctxflintcplx then \(\nu, x \in \mathbb{C}\) is accepted.

See also Wikipedia [1412], MathWorld [1014], NIST [466], BoostMath [121], Mpmath [661].

>>> from xlcalcnet import XComplex
>>> XComplex.HankelH2(10.5, 6.3)
XComplex('5.2359877559829887307E-1')
>>> XComplex.HankelH2(10.5, 6.3)
XComplex('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpc
>>> Gpc.HankelH2(10.5, 6.3)
Gpc('5.2359877559829887307E-1')
>>> Gpc.HankelH2(10.5, 6.3)
Gpc('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; n= 10; x = 30
>>> \mathrm{d}x = dec.hankel2(n, x); mx = mpm.hankel2(n, x); gx = gmp.hankel2(n, x)
>>> fx = fpm.hankel2(n, x); ax = apm.hankel2(n, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax],  aligned=True)
dec: -1.2987689399858876819E-1               - 7.5056702122397113289E-2j
mpm: -1.2987689399858876819e-1               - 7.5056702122397113289e-2j
gmp: -1.2987689399858876819E-01              - 7.5056702122397113289E-02j
fpm: -1.29876893998589E-01                   - 7.50567021223971E-02j
apm: -1.2987689399889094748e-1 (-9.932e-10%) - 7.5056702113900822249e-2 (-2.838e-6%)j

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; n= 10; z = '3 + 4j'
>>> \mathrm{d}z = dec.hankel2(n, z); mz = mpm.hankel2(n, z); gz = gmp.hankel2(n, z)
>>> fz = fpm.hankel2(n, z); az = apm.hankel2(n, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 6.9935943604463042143E+0              - 6.7875888598187273523E+0j
mpm: 6.9935943604463042143e+0              - 6.7875888598187273523e+0j
gmp: 6.9935943604463042143E+00             - 6.7875888598187273523E+00j
fpm: 6.99359436044630E+00                  - 6.78758885981873E+00j
apm: 6.9935943604463042143e+0 (2.132e-18%) - 6.7875888598187273523e+0 (-2.146e-18%)j

Spherical Hankel function of the first kind, \(h^{(1)}_{\nu}(x)\)

ctx.sph_hankel_h1(n, x, scaled=False)

where ctx is math53 or ctxboost.

Returns the spherical Hankel function of the first kind, defined as \(\displaystyle h^{(1)}_{n}(x) = \sqrt{\tfrac{1}{2}\pi} \frac{1}{\sqrt{x}} H^{(1)}_{\nu}(x)= j_{\nu}(x) + i y_{\nu}(x)\).

If scaled is True, then \(\displaystyle h^{(1)e}_{n}(x) = \sqrt{\tfrac{1}{2}\pi} \frac{1}{\sqrt{x}} H^{(1)e}_{\nu}(x)\) is returned.

See also Wikipedia [1422], MathWorld [1019], NIST [471], BoostMath [135].

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

>>> from xlcalcnet import XComplex
>>> XComplex.SphHankelH1(10.5, 6.3)
XComplex('5.2359877559829887307E-1')
>>> XComplex.SphHankelH1(10.5, 6.3)
XComplex('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpc
>>> Gpc.SphHankelH1(10.5, 6.3)
Gpc('5.2359877559829887307E-1')
>>> Gpc.SphHankelH1(10.5, 6.3)
Gpc('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; n= 10; x = 30
>>> \mathrm{d}x = dec.sph_bessel_yn(n, x); mx = mpm.sph_bessel_yn(n, x); gx = gmp.sph_bessel_yn(n, x)
>>> fx = fpm.sph_bessel_yn(n, x); ax = apm.sph_bessel_yn(n, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax],  aligned=True)
dec: 3.121959106475493540775038911812539791138E-2
mpm: 3.121959106475493540775038911812539791138e-2
gmp: 3.121959106475493540775038911812539791138E-02
fpm: 3.12195910647549E-02
apm: 3.121959106475493540775038911819736737226e-2 (1.048e-27%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; n= 10; z = '3 + 4j'
>>> \mathrm{d}z = dec.sph_bessel_yn(n, z); mz = mpm.sph_bessel_yn(n, z); gz = gmp.sph_bessel_yn(n, z)
>>> fz = fpm.sph_bessel_yn(n, z); az = apm.sph_bessel_yn(n, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 1.0803151721461599990E+1              - 1.7336496520486643470E+0j
mpm: 1.0803151721461599990e+1              - 1.7336496520486643470e+0j
gmp: 1.0803151721461599990E+01             - 1.7336496520486643470E+00j
fpm: 1.08031517214616E+01                  - 1.73364965204866E+00j
apm: 1.0803151721461599990e+1 (6.962e-18%) - 1.7336496520486643470e+0 (-3.835e-17%)j

Spherical Hankel function of the second kind, \(h^{(2)}_{\nu}(x)\)

ctx.sph_hankel_h2(n, x, scaled=False)

where ctx is math53 or ctxboost.

Returns the spherical Hankel function of the first kind, defined as \(\displaystyle h^{(2)}_{n}(x) = \sqrt{\tfrac{1}{2}\pi} \frac{1}{\sqrt{x}} H^{(2)}_{\nu}(x)= j_{\nu}(x) - i y_{\nu}(x)\).

If scaled is True, then \(\displaystyle h^{(2)e}_{n}(x) = \sqrt{\tfrac{1}{2}\pi} \frac{1}{\sqrt{x}} H^{(2)e}_{\nu}(x)\) is returned.

See also Wikipedia [1422], MathWorld [1019], NIST [471], BoostMath [135].

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

>>> from xlcalcnet import XComplex
>>> XComplex.SphHankelH1(10.5, 6.3)
XComplex('5.2359877559829887307E-1')
>>> XComplex.SphHankelH1(10.5, 6.3)
XComplex('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpc
>>> Gpc.SphHankelH1(10.5, 6.3)
Gpc('5.2359877559829887307E-1')
>>> Gpc.SphHankelH1(10.5, 6.3)
Gpc('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; n= 10; x = 30
>>> \mathrm{d}x = dec.hankel2(n, x); mx = mpm.hankel2(n, x); gx = gmp.hankel2(n, x)
>>> fx = fpm.hankel2(n, x); ax = apm.hankel2(n, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax],  aligned=True)
dec: -1.2987689399858876819E-1               - 7.5056702122397113289E-2j
mpm: -1.2987689399858876819e-1               - 7.5056702122397113289e-2j
gmp: -1.2987689399858876819E-01              - 7.5056702122397113289E-02j
fpm: -1.29876893998589E-01                   - 7.50567021223971E-02j
apm: -1.2987689399889094748e-1 (-9.932e-10%) - 7.5056702113900822249e-2 (-2.838e-6%)j

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; n= 10; z = '3 + 4j'
>>> \mathrm{d}z = dec.hankel2(n, z); mz = mpm.hankel2(n, z); gz = gmp.hankel2(n, z)
>>> fz = fpm.hankel2(n, z); az = apm.hankel2(n, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 6.9935943604463042143E+0              - 6.7875888598187273523E+0j
mpm: 6.9935943604463042143e+0              - 6.7875888598187273523e+0j
gmp: 6.9935943604463042143E+00             - 6.7875888598187273523E+00j
fpm: 6.99359436044630E+00                  - 6.78758885981873E+00j
apm: 6.9935943604463042143e+0 (2.132e-18%) - 6.7875888598187273523e+0 (-2.146e-18%)j