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Autograd doc note fix (#51661)

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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
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anjali411 2021-02-03 15:05:24 -08:00 committed by Facebook GitHub Bot
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15
docs/source/notes/autograd.rst
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@ -240,13 +240,20 @@ What are complex derivatives?
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 @@ How do these equations translate into complex space :math:`ℂ`?
.. 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}