Autograd doc note fix (#51661)
Summary: Pull Request resolved: https://github.com/pytorch/pytorch/pull/51661
Test Plan: Imported from OSS
Reviewed By: ezyang
Differential Revision: D26230912
Pulled By: anjali411
fbshipit-source-id: 94323d7bce631a4c5781020e9650495461119ede
diff --git a/docs/source/notes/autograd.rst b/docs/source/notes/autograd.rst
index 05b12cc..0590e0dc 100644
--- a/docs/source/notes/autograd.rst
+++ b/docs/source/notes/autograd.rst
@@ -240,13 +240,20 @@
The mathematical definition of complex-differentiability takes the
limit definition of a derivative and generalizes it to operate on
-complex numbers. For a function :math:`f: ℂ → ℂ`, we can write:
+complex numbers. Consider a function :math:`f: ℂ → ℂ`,
+
+ .. math::
+ `f(z=x+yj) = u(x, y) + v(x, y)j`
+
+where :math:`u` and :math:`v` are two variable real valued functions.
+
+Using the derivative definition, we can write:
.. math::
f'(z) = \lim_{h \to 0, h \in C} \frac{f(z+h) - f(z)}{h}
In order for this limit to exist, not only must :math:`u` and :math:`v` must be
-real differentiable (as above), but :math:`f` must also satisfy the Cauchy-Riemann `equations
+real differentiable, but :math:`f` must also satisfy the Cauchy-Riemann `equations
<https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations>`_. In
other words: the limit computed for real and imaginary steps (:math:`h`)
must be equal. This is a more restrictive condition.
@@ -336,8 +343,8 @@
.. math::
\begin{aligned}
- z_{n+1} &= x_n - (s/2) * \frac{\partial L}{\partial x} + 1j * (y_n - (s/2) * \frac{\partial L}{\partial y})
- &= z_n - s * 1/2 * (\frac{\partial L}{\partial x} + j \frac{\partial L}{\partial y})
+ z_{n+1} &= x_n - (s/2) * \frac{\partial L}{\partial x} + 1j * (y_n - (s/2) * \frac{\partial L}{\partial y}) \\
+ &= z_n - s * 1/2 * (\frac{\partial L}{\partial x} + j \frac{\partial L}{\partial y}) \\
&= z_n - s * \frac{\partial L}{\partial z^*}
\end{aligned}
