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Anger, Weber and Lommel functions
===============================================================================
.. _rst_mpm_angerj:
Anger function `\mathbf{J}_{\nu}(x)`
-------------------------------------------------------------------------------
.. method:: CtxFlint.AngerJ(n, z)
Returns the Anger function J. See also Wikipedia :cite:p:`WikipediaFun1046`, MathWorld :cite:p:`WolframFun1046`, NIST :cite:p:`DLMFun1046`.
Gives the Anger function
.. math ::
\mathbf{J}_{\nu}(z) = \frac{1}{\pi}
\int_0^{\pi} \cos(\nu t - z \sin t) dt
which is an entire function of both the parameter `\nu` and
the argument `z`. It solves the inhomogeneous Bessel differential
equation
.. math ::
f''(z) + \frac{1}{z}f'(z) + \left(1-\frac{\nu^2}{z^2}\right) f(z)
= \frac{(z-\nu)}{\pi z^2} \sin(\pi \nu).
We also have
.. math::
:nowrap:
\begin{eqnarray}
\textbf{J}_{\nu}(z) & = & \frac{z}{2} \sin\left(\tfrac{1}{2}\pi\nu \right) {}_1\widetilde{F}_2\left(1; \tfrac{1}{2}(3-\nu), \tfrac{1}{2}(3+\nu); -\frac{z^2}{4} \right) \\
&+& \cos\left(\tfrac{1}{2}\pi\nu \right) {}_1\widetilde{F}_2\left(1; 1-\tfrac{1}{2}\nu, 1+\tfrac{1}{2}\nu; -\frac{z^2}{4} \right) \nonumber
\end{eqnarray}
An example with real input:
.. code-block:: pycon
>>> from mpfebnet import dpm, mpm, gmp, fpm, apm
>>> mpm.dps = 40; n = 10; x = 3
>>> dx = dpm.angerj(n, x); mx = mpm.angerj(n, x); gx = gmp.angerj(n, x)
>>> fx = fpm.angerj(n, x); ax = apm.angerj(n, x)
>>> mpm.show([dx, mx, gx, fx, ax], aligned=True)
dpm: 1.292835164571588377753453080258017074342E-5
mpm: 1.292835164571588377753453080258017074342e-5
gmp: 1.292835164571588377753453080258017074342E-05
fpm: 1.29283516457159E-05
apm: 1.292835164571588377753453080258017065427e-5 (4.139e-35%)
An example with complex input:
.. code-block:: pycon
>>> from mpfebnet import dpm, mpm, gmp, fpm, apm
>>> mpm.dps = 20; n = '10'; z = '3 + 4j'
>>> dz = dpm.angerj(n, z); mz = mpm.angerj(n, z); gz = gmp.angerj(n, z)
>>> fz = fpm.angerj(n, z); az = apm.angerj(n, z)
>>> mpm.show([dz, mz, gz, fz, az], aligned=True)
dpm: -2.4028734611284405858E-3 + 1.9815132418922270634E-3j
mpm: -2.4028734611284405858e-3 + 1.9815132418922270634e-3j
gmp: -2.4028734611284405858E-03 + 1.9815132418922270634E-03j
fpm: -2.40287346112844E-03 + 1.98151324189223E-03j
apm: -2.4028734611284405757e-3 (-4.846e-16%) + 1.9815132418922270691e-3 (4.828e-16%)j
|newpage|
.. _rst_mpm_webere:
Weber function `\mathbf{E}_{\nu}(x)`
-------------------------------------------------------------------------------
.. method:: CtxFlint.WeberE(n, z)
Returns the Weber function E. See also Wikipedia :cite:p:`WikipediaFun1046`, MathWorld :cite:p:`WolframFun1047`, NIST :cite:p:`DLMFun1046`.
Gives the Weber function
.. math ::
\mathbf{E}_{\nu}(z) = \frac{1}{\pi}
\int_0^{\pi} \sin(\nu t - z \sin t) dt
which is an entire function of both the parameter `\nu` and
the argument `z`. It solves the inhomogeneous Bessel differential
equation
.. math ::
f''(z) + \frac{1}{z}f'(z) + \left(1-\frac{\nu^2}{z^2}\right) f(z)
= -\frac{1}{\pi z^2} (z+\nu+(z-\nu)\cos(\pi \nu)).
We also have
.. math::
:nowrap:
\begin{eqnarray}
\textbf{E}_{\nu}(z) & = & \sin\left(\tfrac{1}{2}\pi\nu \right) {}_1\widetilde{F}_2\left(1; \tfrac{1}{2}(2-\nu), \tfrac{1}{2}(2+\nu); -\frac{z^2}{4} \right) \\
&-& \frac{z}{2} \cos\left(\tfrac{1}{2}\pi\nu \right) {}_1\widetilde{F}_2\left(1; \tfrac{1}{2}(3-\nu), \tfrac{1}{2}(3+\nu); -\frac{z^2}{4} \right) \nonumber
\end{eqnarray}
An example with real input:
.. code-block:: pycon
>>> from mpfebnet import dpm, mpm, gmp, fpm, apm
>>> mpm.dps = 40; n = 10; x = 3
>>> dx = dpm.webere(n, x); mx = mpm.webere(n, x); gx = gmp.webere(n, x)
>>> fx = fpm.webere(n, x); ax = apm.webere(n, x)
>>> mpm.show([dx, mx, gx, fx, ax], aligned=True)
dpm: 2.148075016625847487775557330568542804621E-2
mpm: 2.148075016625847487775557330568542804621e-2
gmp: 2.148075016625847487775557330568542804621E-02
fpm: 2.14807501662585E-02
apm: 2.148075016625847487775557330568542804650e-2 (2.929e-36%)
An example with complex input:
.. code-block:: pycon
>>> from mpfebnet import dpm, mpm, gmp, fpm, apm
>>> mpm.dps = 20; n = '10'; z = '3 + 4j'
>>> dz = dpm.webere(n, z); mz = mpm.webere(n, z); gz = gmp.webere(n, z)
>>> fz = fpm.webere(n, z); az = apm.webere(n, z)
>>> mpm.show([dz, mz, gz, fz, az], aligned=True)
dpm: 1.3585638942510994279E-2 + 2.8890345857931466183E-2j
mpm: 1.3585638942510994279e-2 + 2.8890345857931466183e-2j
gmp: 1.3585638942510994279E-02 + 2.8890345857931466183E-02j
fpm: 1.35856389425110E-02 + 2.88903458579315E-02j
apm: 1.3585638942510994281e-2 (3.872e-16%) + 2.8890345857931466164e-2 (2.077e-16%)j
|newpage|
.. _rst_mpm_lommels1:
Lommel function `s_{\mu,\nu}(x) = s^{(1)}_{\mu,\nu}(x)`
-------------------------------------------------------------------------------
.. method:: CtxFlint.LommelS1(mu, nu, z)
Returns the Lommel function S1. See also Wikipedia :cite:p:`WikipediaFun1048`, MathWorld :cite:p:`WolframFun1048`, NIST :cite:p:`DLMFun1048`.
Gives the Lommel function `s_{\mu,\nu}` or `s^{(1)}_{\mu,\nu}`
.. math ::
s_{\mu,\nu}(z) = \frac{z^{\mu+1}}{(\mu-\nu+1)(\mu+\nu+1)}
\,_1F_2\left(1; \frac{\mu-\nu+3}{2}, \frac{\mu+\nu+3}{2};
-\frac{z^2}{4} \right)
which solves the inhomogeneous Bessel equation
.. math ::
z^2 f''(z) + z f'(z) + (z^2-\nu^2) f(z) = z^{\mu+1}.
An integral representation is given by
.. math :: s_{\mu,\nu}(z) = \frac{\pi^2}{2} \left[ Y_{\nu} (z) \! \int_{0}^{z} \!\! t^{\mu} J_{\nu}(t) \, dt - J_\nu (z) \! \int_{0}^{z} \!\! t^{\mu} Y_{\nu}(t) \, dt \right].
A second solution is given by :ref:`lommels2() `.
An example with real input:
.. code-block:: pycon
>>> from mpfebnet import dpm, mpm, gmp, fpm, apm
>>> mpm.dps = 40; mu = '11.3'; nu = '2.7'; x = '0.3'
>>> dx = dpm.lommels1(nu, mu, x); mx = mpm.lommels1(nu, mu, x); gx = gmp.lommels1(nu, mu, x)
>>> fx = fpm.lommels1(nu, mu, x); ax = apm.lommels1(nu, mu, x)
>>> mpm.show([dx, mx, gx, fx, ax])
dpm: -1.020597995063898424938319205823615501143E-4
mpm: -1.020597995063898424938319205823615501143e-4
gmp: -1.020597995063898424938319205823615501143E-04
fpm: -1.02059799506390E-04
apm: -1.020597995063898424938319205823615501143e-4 (-2.293e-37%)
An example with complex input:
.. code-block:: pycon
>>> from mpfebnet import dpm, mpm, gmp, fpm, apm
>>> mpm.dps = 20; nu = '11.0 + 2.0j'; mu = '12.0 + 3.0j'; z = '3.0 + 4.0j'
>>> dz = dpm.lommels1(nu, mu, z); mz = mpm.lommels1(nu, mu, z); gz = gmp.lommels1(nu, mu, z)
>>> fz = fpm.lommels1(nu, mu, z); az = apm.lommels1(nu, mu, z)
>>> mpm.show([dz, mz, gz, fz, az], aligned=True)
dpm: -1.8060086283535056970E+6 + 6.5353506430569991508E+5j
mpm: -1.8060086283535056970e+6 + 6.5353506430569991508e+5j
gmp: -1.8060086283535056970E+06 + 6.5353506430569991508E+05j
fpm: -1.80600862835351E+06 + 6.53535064305700E+05j
apm: -1.8060086283535056970e+6 (-3.443e-19%) + 6.5353506430569991508e+5 (4.077e-19%)j
|newpage|
.. _rst_mpm_lommels2:
Lommel function `S_{\mu,\nu}(x) = s^{(2)}_{\mu,\nu}(x)`
-------------------------------------------------------------------------------
.. method:: CtxFlint.LommelS2(mu, nu, z)
Returns the Lommel function S2. See also Wikipedia :cite:p:`WikipediaFun1048`, MathWorld :cite:p:`WolframFun1048`, NIST :cite:p:`DLMFun1048`.
Gives the second Lommel function `S_{\mu,\nu}` or `s^{(2)}_{\mu,\nu}`
.. math ::
S_{\mu,\nu}(z) = s_{\mu,\nu}(z) + 2^{\mu-1}
\Gamma\left(\tfrac{1}{2}(\mu-\nu+1)\right)
\Gamma\left(\tfrac{1}{2}(\mu+\nu+1)\right) \times
\left[\sin(\tfrac{1}{2}(\mu-\nu)\pi) J_{\nu}(z) -
\cos(\tfrac{1}{2}(\mu-\nu)\pi) Y_{\nu}(z)
\right]
which solves the same differential equation as :ref:`lommels1() `.
An example with real input:
.. code-block:: pycon
>>> from mpfebnet import dpm, mpm, gmp, fpm, apm
>>> mpm.dps = 40; mu = '11.3'; nu = '2.7'; x = '0.3'
>>> dx = dpm.lommels2(nu, mu, x); mx = mpm.lommels2(nu, mu, x); gx = gmp.lommels2(nu, mu, x)
>>> fx = fpm.lommels2(nu, mu, x); ax = apm.lommels2(nu, mu, x)
>>> mpm.show([dx, mx, gx, fx, ax])
dpm: 5.148372921395779596423917787112324661480E+18
mpm: 5.148372921395779596423917787112324661480e+18
gmp: 5.148372921395779596423917787112324661480E+18
fpm: 5.14837292139580E+18
apm: 5.148372921395779596423917787112320930634e+18 (7.037e-31%)
An example with complex input:
.. code-block:: pycon
>>> from mpfebnet import dpm, mpm, gmp, fpm, apm
>>> mpm.dps = 20; nu = '11.0 + 2.0j'; mu = '12.0 + 3.0j'; z = '3.0 + 4.0j'
>>> dz = dpm.lommels2(nu, mu, z); mz = mpm.lommels2(nu, mu, z); gz = gmp.lommels2(nu, mu, z)
>>> fz = fpm.lommels2(nu, mu, z); az = apm.lommels2(nu, mu, z)
>>> mpm.show([dz, mz, gz, fz, az], aligned=True)
dpm: -5.9447048505419644513E+13 + 2.6440996022817513605E+14j
mpm: -5.9447048505419644513e+13 + 2.6440996022817513605e+14j
gmp: -5.9447048505419644513E+13 + 2.6440996022817513605E+14j
fpm: -5.94470485054196E+13 + 2.64409960228175E+14j
apm: -5.9447048505428070102e+13 (-1.1e-10%) + 2.6440996022816726372e+14 (3.001e-11%)j