 II**

    There are several definitions of the DST-II; we use the following for
    ``norm=None``. DST-II assumes the input is odd around `n=-1/2` and
    `n=N-1/2`; the output is odd around :math:`k=-1` and even around `k=N-1`

    .. math::

        y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(k+1)(2n+1)}{2N}\right)

    if ``norm='ortho'``, ``y[k]`` is multiplied by a scaling factor ``f``

    .. math::

        f = \begin{cases}
        \sqrt{\frac{1}{4N}} & \text{if }k = 0, \\
        \sqrt{\frac{1}{2N}} & \text{otherwise} \end{cases}

    **Type III**

    There are several definitions of the DST-III, we use the following (for
    ``norm=None``). DST-III assumes the input is odd around `n=-1` and even
    around `n=N-1`

    .. math::

        y_k = (-1)^k x_{N-1} + 2 \sum_{n=0}^{N-2} x_n \sin\left(
        \frac{\pi(2k+1)(n+1)}{2N}\right)

    The (unnormalized) DST-III is the inverse of the (unnormalized) DST-II, up
    to a factor `2N`. The orthonormalized DST-III is exactly the inverse of the
    orthonormalized DST-II.

    .. versionadded:: 0.11.0

    **Type IV**

    There are several definitions of the DST-IV, we use the following (for
    ``norm=None``). DST-IV assumes the input is odd around `n=-0.5` and even
    around `n=N-0.5`

    .. math::

        y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(2k+1)(2n+1)}{4N}\right)

    The (unnormalized) DST-IV is its own inverse, up to a factor `2N`. The
    orthonormalized DST-IV is exactly its own inverse.

    .. versionadded:: 1.2.0
       Support for DST-IV.

    References
    ----------
    .. [1] Wikipedia, "Discrete sine transform",
           https://en.wikipedia.org/wiki/Discrete_sine_transform

    )