Voigt-Hjerting Function and Voigt Profile
---------------------------------------------------
*Update: May 21/2021, Hajime Kawahara*


Voigt-Hjerting Function
=============================

The Voigt-Hjerting function is the real part of `the Faddeeva function <https://en.wikipedia.org/wiki/Faddeeva_function>`_, defined as

:math:`H(x,a) = \frac{a}{\pi}`
:math:`\int_{-\infty}^{\infty}`
:math:`\frac{e^{-y^2}}{(x-y)^2 + a^2} dy` .

In exojax,
*hjert*
provides the Voigt-Hjerting function.

.. code:: ipython3

    from exojax.spec import hjert
    hjert(1.0,1.0)


.. parsed-literal::

    DeviceArray(0.3047442, dtype=float32)



We can differentiate the Voigt-Hjerting function by :math:`x`.
:math:`\partial_x H(x,a)` is given by

.. code:: ipython3

    from jax import grad
    dhjert_dx=grad(hjert,argnums=0)
    dhjert_dx(1.0,1.0)




.. parsed-literal::

    DeviceArray(-0.1930505, dtype=float32)



hjert is compatible to JAX. So, when you want to use array as input, you
need to wrap it by `jax.vmap <https://jax.readthedocs.io/en/latest/jax.html#jax.vmap>`_.

.. code:: ipython3

    import jax.numpy as jnp
    from jax import vmap
    import matplotlib.pyplot as plt
    
    #input vector
    x=jnp.linspace(-5,5,100)
    
    #vectorized hjert H(x,a)
    vhjert=vmap(hjert,(0,None),0)
    
    #vectroized dH(x,a)/dx
    vdhjert_dx=vmap(dhjert_dx,(0,None),0)
    
    plt.plot(x, vhjert(x,1.0),label="$H(x,a)$")
    plt.plot(x, vdhjert_dx(x,1.0),label="$\\partial_x H(x,a)$")
    plt.legend()


.. image:: hjerting/output_5_1.png

Voigt Profile
===================

Using the Voigt-Hjerting function, the Voigt profile is expressed as  

:math:`V(\nu, \beta, \gamma_L) = \frac{1}{\sqrt{2 \pi} \beta} H\left( \frac{\nu}{\sqrt{2} \beta},\frac{\gamma_L}{\sqrt{2} \beta} \right)`

:math:`V(\nu, \beta, \gamma_L)` is a convolution of a Gaussian
with a STD of :math:`\beta` and a Lorentian with a gamma parameter of
:math:`\gamma_L`.

.. code:: ipython3

    from exojax.spec import voigt
    import jax.numpy as jnp
    import matplotlib.pyplot as plt
      
.. code:: ipython3

    nu=jnp.linspace(-10,10,100)
    plt.plot(nu, voigt(nu,1.0,2.0)) #beta=1.0, gamma_L=2.0

.. image:: hjerting/output_3_1.png

See " :doc:`../tutorials/voigt_function` " for the tutorial of the Voigt profile.