Add missing imaginary unit

qutil/signal_processing/fourier_space.py

@@ -115,16 +115,25 @@ def Id(x: _S, f: _T, *_, **__) -> Tuple[_S, _T]:
 
 def derivative(x, f, deriv_order: int = 0, overwrite_x: bool = False,
                **_) -> Tuple[np.ndarray, np.ndarray]:
-    """Perform (anti-)derivatives.
+    r"""Compute the (anti-)derivative.
+
+    The sign convention is according to the following defintion of the
+    Fourier transform:
+
+    .. math::
+        \hat{f}(\omega) &= \int_{-\infty}^\infty\mathrm{d}t
+            e^{-i\omega t} f(t) \\
+        f(t) &= \int_{-\infty}^\infty\frac{\mathrm{d}\omega}{2\pi}
+            e^{i\omega t} \hat{f}(\omega)
 
     .. note::
-        For negative antiderivatives, the zero-frequency component is
-        set to zero (due to zero-division).
+        For negative ``deriv_order`` (antiderivatives), the
+        zero-frequency component is set to zero (due to zero-division).
 
     Parameters
     ----------
     x : array_like
-        The data to be filtered.
+        Target data.
     f : array_like
         Frequencies corresponding to the last axis of `x`.
     deriv_order : int
@@ -136,7 +145,7 @@ def derivative(x, f, deriv_order: int = 0, overwrite_x: bool = False,
     x = np.array(x, copy=not overwrite_x)
     f = np.asanyarray(f)
     with np.errstate(invalid='ignore', divide='ignore'):
-        x *= (2*np.pi*f)**deriv_order
+        x *= (2j*np.pi*f)**deriv_order
     if deriv_order < 0:
         x[..., (f == 0).nonzero()] = 0
     return x, f