The Voigt-Hjerting is defined as

$ H(x,a) = :raw-latex:`\frac{a}{\pi}`
:raw-latex:`\int`\_{-:raw-latex:`\infty`}^{:raw-latex:`\infty`}
:raw-latex:`\frac{e^{-y^2}}{(x-y)^2 + a^2}` dy.$

In exojax, hjert is the Voigt-Hjerting function.

.. code:: ipython3

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




.. parsed-literal::

    Array(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::

    Array(-0.19305044, dtype=float32, weak_type=True)



hjert is compatible to JAX. So, when you want to use array as input, you
need to wrap it by 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()




.. parsed-literal::

    <matplotlib.legend.Legend at 0x7f062c293820>




.. image:: hjerting_files/hjerting_5_1.png