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|newpage| Hankel functions =============================================================================== Hankel function of the first kind `H^{(1)}_{\nu}(x)` ------------------------------------------------------------------------------- .. method:: 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 :cite:p:`WikipediaFun142`, MathWorld :cite:p:`WolframFun142a`, NIST :cite:p:`DLMFun142`, BoostMath :cite:p:`BoostFun142`, Mpmath :cite:p:`MpmathFun142a`. .. code-block:: pycon >>> 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 .. code-block:: pycon >>> 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: .. code-block:: pycon >>> 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: .. code-block:: pycon >>> 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 |newpage| Hankel function of the second kind `H^{(2)}_{\nu}(x)` ------------------------------------------------------------------------------- .. method:: 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 :cite:p:`WikipediaFun142`, MathWorld :cite:p:`WolframFun142b`, NIST :cite:p:`DLMFun142`, BoostMath :cite:p:`BoostFun142`, Mpmath :cite:p:`MpmathFun142b`. .. code-block:: pycon >>> 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 .. code-block:: pycon >>> 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: .. code-block:: pycon >>> 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: .. code-block:: pycon >>> 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 |newpage| Spherical Hankel function of the first kind, `h^{(1)}_{\nu}(x)` ------------------------------------------------------------------------------- .. method:: 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 :cite:p:`WikipediaFun144`, MathWorld :cite:p:`WolframFun144`, NIST :cite:p:`DLMFun144`, BoostMath :cite:p:`BoostFun144`. 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. .. code-block:: pycon >>> 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 .. code-block:: pycon >>> 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: .. code-block:: pycon >>> 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: .. code-block:: pycon >>> 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 |newpage| Spherical Hankel function of the second kind, `h^{(2)}_{\nu}(x)` ------------------------------------------------------------------------------- .. method:: 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 :cite:p:`WikipediaFun144`, MathWorld :cite:p:`WolframFun144`, NIST :cite:p:`DLMFun144`, BoostMath :cite:p:`BoostFun144`. 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. .. code-block:: pycon >>> 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 .. code-block:: pycon >>> 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: .. code-block:: pycon >>> 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: .. code-block:: pycon >>> 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