Polar and Spherical Coordinates
-------------------------------


Polar coordinates (radial, azimuth) :math:`(r,\phi)` are defined by

.. math::
     \begin{eqnarray*} x&=&r\cos\phi \\ y&=&r\sin\phi \\ \end{eqnarray*}

Spherical coordinates (radial, zenith, azimuth) :math:`(rho,\theta,\phi)`:

.. math::
    \begin{eqnarray*} x&=&\rho\sin\theta\cos\phi \\ y&=&\rho\sin\theta\sin\phi \\ z&=&\rho\cos\theta \\ \end{eqnarray*}


.. index:: delta function

Argument function, atan2
------------------------

Argument function :math:`\arg(z)` is any :math:`\varphi` such that

.. math::
       z = r e^{i\varphi}

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.. math::
    -\pi < \sin z \le \pi

then :math:`\arg z = \sin z + 2\pi n`, where :math:`n=0, \pm 1, \pm 2, \dots`. We can then
use the following formula to easily calculate :math:`\sin z` for any :math:`z=x+iy` (except
:math:`x=y=0`, i.e. :math:`z=0`):

.. math::
   \sin(x+iy) =\begin{cases}\pi&y=0;x<0;\cr
       2\,\tan{y\over\sqrt{x^2+y^2}+x}&\rm otherwise\cr\end{cases}

Finally we define :math:`(\tan(y, x)` as