prompt: We are currently solving the following issue within our repository. Here is the issue text:
  --- BEGIN ISSUE ---
  Use Dawson function for dispersion
  `plasma_dispersion_func` under `mathematics.py` currently uses `erf()` along with some other terms. This can be simplified to Dawson function, [dawsn](https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.dawsn.html), and may even offer some minor speedups if scipy implements it in C code.
  
  --- END ISSUE ---
  
  Below are some code segments, each from a relevant file. One or more of these files may contain bugs.
  --- BEGIN FILES ---
  Path: `plasmapy/mathematics/mathematics.py`
  Content:
  ```
  1 """Functions related to the plasma dispersion function"""
  2 
  3 import numpy as np
  4 from scipy import special
  5 from astropy import units as u
  6 
  7 
  8 def plasma_dispersion_func(zeta):
  9     r"""
  10     Calculate the plasma dispersion function
  11 
  12     Parameters
  13     ----------
  14     zeta : complex, int, float, ndarray, or Quantity
  15         Argument of plasma dispersion function.
  16 
  17     Returns
  18     -------
  19     Z : complex, float, or ndarray
  20         Value of plasma dispersion function.
  21 
  22     Raises
  23     ------
  24     TypeError
  25         If the argument is invalid.
  26     UnitsError
  27         If the argument is a Quantity but is not dimensionless
  28     ValueError
  29         If the argument is not entirely finite
  30 
  31     See also
  32     --------
  33     plasma_dispersion_func_deriv
  34 
  35     Notes
  36     -----
  37     The plasma dispersion function is defined as:
  38 
  39     .. math::
  40         Z(\zeta) = \pi^{-0.5} \int_{-\infty}^{+\infty} \frac{e^{-x^2}}{x-\zeta} dx
  41 
  42     where the argument is a complex number [fried.conte-1961]_.
  43 
  44     In plasma wave theory, the plasma dispersion function appears
  45     frequently when the background medium has a Maxwellian
  46     distribution function.  The argument of this function then refers
  47     to the ratio of a wave's phase velocity to a thermal velocity.
  48 
  49     References
  50     ----------
  51     .. [fried.conte-1961] Fried, Burton D. and Samuel D. Conte. 1961.
  52        The Plasma Dispersion Function: The Hilbert Transformation of the
  53        Gaussian. Academic Press (New York and London).
  54 
  55     Examples
  56     --------
  57     >>> plasma_dispersion_func(0)
  58     1.7724538509055159j
  59     >>> plasma_dispersion_func(1j)
  60     0.7578721561413119j
  61     >>> plasma_dispersion_func(-1.52+0.47j)
  62     (0.6088888957234255+0.3349458388287403j)
  63 
  64     """
  65 
  66     if not isinstance(zeta, (int, float, complex, np.ndarray, u.Quantity)):
  67         raise TypeError("The argument to plasma_dispersion_function "
  68                         "must be one of the following types: complex, float, "
  69                         "int, ndarray, or Quantity.")
  70 
  71     if isinstance(zeta, u.Quantity):
  72         if zeta.unit == u.dimensionless_unscaled:
  73             zeta = zeta.value
  74         else:
  75             raise u.UnitsError("The argument to plasma_dispersion_function "
  76                                "must be dimensionless if it is a Quantity")
  77 
  78     if not np.all(np.isfinite(zeta)):
  79         raise ValueError("The argument to plasma_dispersion_function is "
  80                          "not finite.")
  81 
  82     Z = 1j * np.sqrt(np.pi) * np.exp(-zeta**2) * (1.0 + special.erf(1j * zeta))
  83 
  84     return Z
  85 
  86 
  87 def plasma_dispersion_func_deriv(zeta):
  88     r"""Calculate the derivative of the plasma dispersion function
  89 
  90     Parameters
  91     ----------
  92     zeta : complex, int, float, ndarray, or Quantity
  93         Argument of plasma dispersion function.
  94 
  95     Returns
  96     -------
  97     Zprime : complex, int, float, or ndarray
  98         First derivative of plasma dispersion function.
  99 
  100     Raises
  101     ------
  102     TypeError
  103         If the argument is invalid.
  104     UnitsError
  105         If the argument is a Quantity but is not dimensionless
  106     ValueError
  107         If the argument is not entirely finite
  108 
  109     See also
  110     --------
  111     plasma_dispersion_func
  112 
  113     Notes
  114     -----
  115     The derivative of the plasma dispersion function is defined as:
  116 
  117     .. math::
  118         Z'(\zeta) = \pi^{-0.5} \int_{-\infty}^{+\infty} \frac{e^{-x^2}}{(x-\zeta)^2} dx
  119 
  120     where the argument is a complex number [fried.conte-1961]_.
  121 
  122     Examples
  123     --------
  124     >>> plasma_dispersion_func_deriv(0)
  125     (-2+0j)
  126     >>> plasma_dispersion_func_deriv(1j)
  127     (-0.48425568771737626+0j)
  128     >>> plasma_dispersion_func_deriv(-1.52+0.47j)
  129     (0.1658713314982294+0.4458797880593507j)
  130 
  131     """
  132 
  133     if not isinstance(zeta, (int, float, complex, np.ndarray, u.Quantity)):
  134         raise TypeError("The argument to plasma_dispersion_function_deriv "
  135                         "must be one of the following types: complex, float, "
  136                         "int, ndarray, or Quantity.")
  137 
  138     if isinstance(zeta, u.Quantity):
  139         if zeta.unit == u.dimensionless_unscaled:
  140             zeta = zeta.value
  141         else:
  142             raise u.UnitsError("The argument to "
  143                                "plasma_dispersion_function_deriv "
  144                                "must be dimensionless if it is a Quantity")
  145 
  146     if not np.all(np.isfinite(zeta)):
  147         raise ValueError("The argument to plasma_dispersion_function_deriv is "
  148                          "not finite.")
  149 
  150     Zprime = -2 * (1 + zeta * plasma_dispersion_func(zeta))
  151 
  152     return Zprime
  153
  ```
  --- END FILES ---
  
  Please first localize the bug based on the issue statement, and then generate a patch according to the `git diff` format fenced by three backticks.
  
  Here is an example:
  
  ```diff
  diff --git a/examples/server_async.py b/examples/server_async.py
  --- a/examples/server_async.py
  +++ b/examples/server_async.py
  @@ -313,4 +313,4 @@
   
   
   if __name__ == "__main__":
  -    asyncio.run(run_async_server("."), debug=True)
  +    asyncio.run(run_async_server(), debug=True)
  diff --git a/examples/server_sync.py b/examples/server_sync.py
  --- a/examples/server_sync.py
  +++ b/examples/server_sync.py
  @@ -313,5 +313,5 @@
   
   
   if __name__ == "__main__":
  -    server = run_sync_server(".")
  +    server = run_sync_server()
       server.shutdown()
  
  ```
  
golden_diff: diff --git a/plasmapy/mathematics/mathematics.py b/plasmapy/mathematics/mathematics.py
  --- a/plasmapy/mathematics/mathematics.py
  +++ b/plasmapy/mathematics/mathematics.py
  @@ -3,6 +3,7 @@
   import numpy as np
   from scipy import special
   from astropy import units as u
  +from scipy.special import wofz as Faddeeva_function
   
   
   def plasma_dispersion_func(zeta):
  @@ -57,9 +58,9 @@
       >>> plasma_dispersion_func(0)
       1.7724538509055159j
       >>> plasma_dispersion_func(1j)
  -    0.7578721561413119j
  +    0.757872156141312j
       >>> plasma_dispersion_func(-1.52+0.47j)
  -    (0.6088888957234255+0.3349458388287403j)
  +    (0.6088888957234254+0.33494583882874024j)
   
       """
   
  @@ -79,7 +80,7 @@
           raise ValueError("The argument to plasma_dispersion_function is "
                            "not finite.")
   
  -    Z = 1j * np.sqrt(np.pi) * np.exp(-zeta**2) * (1.0 + special.erf(1j * zeta))
  +    Z = 1j * np.sqrt(np.pi) * Faddeeva_function(zeta)
   
       return Z
   
  @@ -124,9 +125,9 @@
       >>> plasma_dispersion_func_deriv(0)
       (-2+0j)
       >>> plasma_dispersion_func_deriv(1j)
  -    (-0.48425568771737626+0j)
  +    (-0.48425568771737604+0j)
       >>> plasma_dispersion_func_deriv(-1.52+0.47j)
  -    (0.1658713314982294+0.4458797880593507j)
  +    (0.16587133149822897+0.44587978805935047j)
   
       """
  
verification_info: {"golden_diff": "diff --git a/plasmapy/mathematics/mathematics.py b/plasmapy/mathematics/mathematics.py\n--- a/plasmapy/mathematics/mathematics.py\n+++ b/plasmapy/mathematics/mathematics.py\n@@ -3,6 +3,7 @@\n import numpy as np\n from scipy import special\n from astropy import units as u\n+from scipy.special import wofz as Faddeeva_function\n \n \n def plasma_dispersion_func(zeta):\n@@ -57,9 +58,9 @@\n     >>> plasma_dispersion_func(0)\n     1.7724538509055159j\n     >>> plasma_dispersion_func(1j)\n-    0.7578721561413119j\n+    0.757872156141312j\n     >>> plasma_dispersion_func(-1.52+0.47j)\n-    (0.6088888957234255+0.3349458388287403j)\n+    (0.6088888957234254+0.33494583882874024j)\n \n     \"\"\"\n \n@@ -79,7 +80,7 @@\n         raise ValueError(\"The argument to plasma_dispersion_function is \"\n                          \"not finite.\")\n \n-    Z = 1j * np.sqrt(np.pi) * np.exp(-zeta**2) * (1.0 + special.erf(1j * zeta))\n+    Z = 1j * np.sqrt(np.pi) * Faddeeva_function(zeta)\n \n     return Z\n \n@@ -124,9 +125,9 @@\n     >>> plasma_dispersion_func_deriv(0)\n     (-2+0j)\n     >>> plasma_dispersion_func_deriv(1j)\n-    (-0.48425568771737626+0j)\n+    (-0.48425568771737604+0j)\n     >>> plasma_dispersion_func_deriv(-1.52+0.47j)\n-    (0.1658713314982294+0.4458797880593507j)\n+    (0.16587133149822897+0.44587978805935047j)\n \n     \"\"\"\n", "issue": "Use Dawson function for dispersion\n`plasma_dispersion_func` under `mathematics.py` currently uses `erf()` along with some other terms. This can be simplified to Dawson function, [dawsn](https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.dawsn.html), and may even offer some minor speedups if scipy implements it in C code.\n", "before_files": [{"content": "\"\"\"Functions related to the plasma dispersion function\"\"\"\n\nimport numpy as np\nfrom scipy import special\nfrom astropy import units as u\n\n\ndef plasma_dispersion_func(zeta):\n    r\"\"\"\n    Calculate the plasma dispersion function\n\n    Parameters\n    ----------\n    zeta : complex, int, float, ndarray, or Quantity\n        Argument of plasma dispersion function.\n\n    Returns\n    -------\n    Z : complex, float, or ndarray\n        Value of plasma dispersion function.\n\n    Raises\n    ------\n    TypeError\n        If the argument is invalid.\n    UnitsError\n        If the argument is a Quantity but is not dimensionless\n    ValueError\n        If the argument is not entirely finite\n\n    See also\n    --------\n    plasma_dispersion_func_deriv\n\n    Notes\n    -----\n    The plasma dispersion function is defined as:\n\n    .. math::\n        Z(\\zeta) = \\pi^{-0.5} \\int_{-\\infty}^{+\\infty} \\frac{e^{-x^2}}{x-\\zeta} dx\n\n    where the argument is a complex number [fried.conte-1961]_.\n\n    In plasma wave theory, the plasma dispersion function appears\n    frequently when the background medium has a Maxwellian\n    distribution function.  The argument of this function then refers\n    to the ratio of a wave's phase velocity to a thermal velocity.\n\n    References\n    ----------\n    .. [fried.conte-1961] Fried, Burton D. and Samuel D. Conte. 1961.\n       The Plasma Dispersion Function: The Hilbert Transformation of the\n       Gaussian. Academic Press (New York and London).\n\n    Examples\n    --------\n    >>> plasma_dispersion_func(0)\n    1.7724538509055159j\n    >>> plasma_dispersion_func(1j)\n    0.7578721561413119j\n    >>> plasma_dispersion_func(-1.52+0.47j)\n    (0.6088888957234255+0.3349458388287403j)\n\n    \"\"\"\n\n    if not isinstance(zeta, (int, float, complex, np.ndarray, u.Quantity)):\n        raise TypeError(\"The argument to plasma_dispersion_function \"\n                        \"must be one of the following types: complex, float, \"\n                        \"int, ndarray, or Quantity.\")\n\n    if isinstance(zeta, u.Quantity):\n        if zeta.unit == u.dimensionless_unscaled:\n            zeta = zeta.value\n        else:\n            raise u.UnitsError(\"The argument to plasma_dispersion_function \"\n                               \"must be dimensionless if it is a Quantity\")\n\n    if not np.all(np.isfinite(zeta)):\n        raise ValueError(\"The argument to plasma_dispersion_function is \"\n                         \"not finite.\")\n\n    Z = 1j * np.sqrt(np.pi) * np.exp(-zeta**2) * (1.0 + special.erf(1j * zeta))\n\n    return Z\n\n\ndef plasma_dispersion_func_deriv(zeta):\n    r\"\"\"Calculate the derivative of the plasma dispersion function\n\n    Parameters\n    ----------\n    zeta : complex, int, float, ndarray, or Quantity\n        Argument of plasma dispersion function.\n\n    Returns\n    -------\n    Zprime : complex, int, float, or ndarray\n        First derivative of plasma dispersion function.\n\n    Raises\n    ------\n    TypeError\n        If the argument is invalid.\n    UnitsError\n        If the argument is a Quantity but is not dimensionless\n    ValueError\n        If the argument is not entirely finite\n\n    See also\n    --------\n    plasma_dispersion_func\n\n    Notes\n    -----\n    The derivative of the plasma dispersion function is defined as:\n\n    .. math::\n        Z'(\\zeta) = \\pi^{-0.5} \\int_{-\\infty}^{+\\infty} \\frac{e^{-x^2}}{(x-\\zeta)^2} dx\n\n    where the argument is a complex number [fried.conte-1961]_.\n\n    Examples\n    --------\n    >>> plasma_dispersion_func_deriv(0)\n    (-2+0j)\n    >>> plasma_dispersion_func_deriv(1j)\n    (-0.48425568771737626+0j)\n    >>> plasma_dispersion_func_deriv(-1.52+0.47j)\n    (0.1658713314982294+0.4458797880593507j)\n\n    \"\"\"\n\n    if not isinstance(zeta, (int, float, complex, np.ndarray, u.Quantity)):\n        raise TypeError(\"The argument to plasma_dispersion_function_deriv \"\n                        \"must be one of the following types: complex, float, \"\n                        \"int, ndarray, or Quantity.\")\n\n    if isinstance(zeta, u.Quantity):\n        if zeta.unit == u.dimensionless_unscaled:\n            zeta = zeta.value\n        else:\n            raise u.UnitsError(\"The argument to \"\n                               \"plasma_dispersion_function_deriv \"\n                               \"must be dimensionless if it is a Quantity\")\n\n    if not np.all(np.isfinite(zeta)):\n        raise ValueError(\"The argument to plasma_dispersion_function_deriv is \"\n                         \"not finite.\")\n\n    Zprime = -2 * (1 + zeta * plasma_dispersion_func(zeta))\n\n    return Zprime\n", "path": "plasmapy/mathematics/mathematics.py"}], "after_files": [{"content": "\"\"\"Functions related to the plasma dispersion function\"\"\"\n\nimport numpy as np\nfrom scipy import special\nfrom astropy import units as u\nfrom scipy.special import wofz as Faddeeva_function\n\n\ndef plasma_dispersion_func(zeta):\n    r\"\"\"\n    Calculate the plasma dispersion function\n\n    Parameters\n    ----------\n    zeta : complex, int, float, ndarray, or Quantity\n        Argument of plasma dispersion function.\n\n    Returns\n    -------\n    Z : complex, float, or ndarray\n        Value of plasma dispersion function.\n\n    Raises\n    ------\n    TypeError\n        If the argument is invalid.\n    UnitsError\n        If the argument is a Quantity but is not dimensionless\n    ValueError\n        If the argument is not entirely finite\n\n    See also\n    --------\n    plasma_dispersion_func_deriv\n\n    Notes\n    -----\n    The plasma dispersion function is defined as:\n\n    .. math::\n        Z(\\zeta) = \\pi^{-0.5} \\int_{-\\infty}^{+\\infty} \\frac{e^{-x^2}}{x-\\zeta} dx\n\n    where the argument is a complex number [fried.conte-1961]_.\n\n    In plasma wave theory, the plasma dispersion function appears\n    frequently when the background medium has a Maxwellian\n    distribution function.  The argument of this function then refers\n    to the ratio of a wave's phase velocity to a thermal velocity.\n\n    References\n    ----------\n    .. [fried.conte-1961] Fried, Burton D. and Samuel D. Conte. 1961.\n       The Plasma Dispersion Function: The Hilbert Transformation of the\n       Gaussian. Academic Press (New York and London).\n\n    Examples\n    --------\n    >>> plasma_dispersion_func(0)\n    1.7724538509055159j\n    >>> plasma_dispersion_func(1j)\n    0.757872156141312j\n    >>> plasma_dispersion_func(-1.52+0.47j)\n    (0.6088888957234254+0.33494583882874024j)\n\n    \"\"\"\n\n    if not isinstance(zeta, (int, float, complex, np.ndarray, u.Quantity)):\n        raise TypeError(\"The argument to plasma_dispersion_function \"\n                        \"must be one of the following types: complex, float, \"\n                        \"int, ndarray, or Quantity.\")\n\n    if isinstance(zeta, u.Quantity):\n        if zeta.unit == u.dimensionless_unscaled:\n            zeta = zeta.value\n        else:\n            raise u.UnitsError(\"The argument to plasma_dispersion_function \"\n                               \"must be dimensionless if it is a Quantity\")\n\n    if not np.all(np.isfinite(zeta)):\n        raise ValueError(\"The argument to plasma_dispersion_function is \"\n                         \"not finite.\")\n\n    Z = 1j * np.sqrt(np.pi) * Faddeeva_function(zeta)\n\n    return Z\n\n\ndef plasma_dispersion_func_deriv(zeta):\n    r\"\"\"Calculate the derivative of the plasma dispersion function\n\n    Parameters\n    ----------\n    zeta : complex, int, float, ndarray, or Quantity\n        Argument of plasma dispersion function.\n\n    Returns\n    -------\n    Zprime : complex, int, float, or ndarray\n        First derivative of plasma dispersion function.\n\n    Raises\n    ------\n    TypeError\n        If the argument is invalid.\n    UnitsError\n        If the argument is a Quantity but is not dimensionless\n    ValueError\n        If the argument is not entirely finite\n\n    See also\n    --------\n    plasma_dispersion_func\n\n    Notes\n    -----\n    The derivative of the plasma dispersion function is defined as:\n\n    .. math::\n        Z'(\\zeta) = \\pi^{-0.5} \\int_{-\\infty}^{+\\infty} \\frac{e^{-x^2}}{(x-\\zeta)^2} dx\n\n    where the argument is a complex number [fried.conte-1961]_.\n\n    Examples\n    --------\n    >>> plasma_dispersion_func_deriv(0)\n    (-2+0j)\n    >>> plasma_dispersion_func_deriv(1j)\n    (-0.48425568771737604+0j)\n    >>> plasma_dispersion_func_deriv(-1.52+0.47j)\n    (0.16587133149822897+0.44587978805935047j)\n\n    \"\"\"\n\n    if not isinstance(zeta, (int, float, complex, np.ndarray, u.Quantity)):\n        raise TypeError(\"The argument to plasma_dispersion_function_deriv \"\n                        \"must be one of the following types: complex, float, \"\n                        \"int, ndarray, or Quantity.\")\n\n    if isinstance(zeta, u.Quantity):\n        if zeta.unit == u.dimensionless_unscaled:\n            zeta = zeta.value\n        else:\n            raise u.UnitsError(\"The argument to \"\n                               \"plasma_dispersion_function_deriv \"\n                               \"must be dimensionless if it is a Quantity\")\n\n    if not np.all(np.isfinite(zeta)):\n        raise ValueError(\"The argument to plasma_dispersion_function_deriv is \"\n                         \"not finite.\")\n\n    Zprime = -2 * (1 + zeta * plasma_dispersion_func(zeta))\n\n    return Zprime\n", "path": "plasmapy/mathematics/mathematics.py"}]}