\ifdefined\inmaster\else\def\subonly{\jobname}\input{complex}\fi

\chapter{Complex differential forms}

So far, we have discussed only \emph{real} differential forms.
We now apply these to the study of complex numbers and complex differentials.

\section{Complex differential forms}
\label{sec:complex-forms}

There is no problem with allowing the components of a differential form to be \emph{complex-valued}.
Thus, for instance, we may consider the differential 1-form $e^{ix} \,\dx + (y-ix^2)\,\dy$, or the differential 2-form $(x^2- i y^2)\,\dx\wedge\dy$.
Note that the ambient space is still (one-, two-, or three-dimensional) \emph{real} space, with the coordinates such as $x$ and $y$ being real numbers.

However, let us now restrict to considering such forms in \emph{two} dimensions, with real coordinates $x$ and $y$.
In this case, as we have seen, we can also identify the plane with the set of complex numbers, with complex coordinate $z = x+iy$.
If we regard $z$ as a complex-valued function of the two real coordinates $x$ and $y$, i.e.\ $z(x,y) = x+iy$, then its differential is a complex-valued 1-form:
\[ \dz = \dx + i \,\dy. \]
Similarly, its conjugate $\zbar$ is also a complex-valued function of $x$ and $y$, namely $\zbar(x,y) = x-iy$, and its differential is
\[ \dzbar = \dx - i\, \dy. \]

Now note that just as
\[ x = \half(z+\zbar) \qquad\text{and}\qquad y = \half(z-\zbar) \]
we have
\[ \dx = \half(\dz+\dzbar) \qquad\text{and}\qquad \dy = \halfi(\dz-\dzbar) \]
Therefore, any complex-valued differential form can be written in terms of $\dz$ and $\dzbar$ instead of $\dx$ and $\dy$.
Specifically, if we have a 1-form $f(x,y) \,\dx + g(x,y) \,\dy$, then we can write it as
\begin{align*}
  f(x,y) \,\dx + g(x,y) \,\dy
  &= f(x,y)\cdot\half(\dz+\dzbar) +  g(x,y) \cdot\halfi(\dz-\dzbar)\\
  &= \half (f(x,y) -i\, g(x,y))\,\dz + \half (f(x,y) +i\, g(x,y))\,\dzbar.
\end{align*}
We also have
\begin{align*}
  \dz \wedge\dzbar &= (\dx + i \,\dy) \wedge (\dx - i \,\dy)\\
  &= \dx\wedge\dx + i\,\dy\wedge\dx - i\,\dx\wedge\dy + \dy\wedge\dy\\
  &= -2i\, \dx\wedge\dy.
\end{align*}
Therefore, a 2-form $A\,\dx\wedge\dy$ can equivalently be written as $-\halfi A\, \dz\wedge\dzbar$.

\section{Complex differentials}
\label{sec:complex-differentials}

Now let us consider the differential as it acts on complex forms.
Suppose $f$ is a complex-valued 0-form, i.e.\ a function.
Then we have
\begin{align*}
  \df &= \pder f x \, \dx + \pder f y\,\dy\\
  &= \half (\pder f x - i\, \pder f y)\,\dz + \half (\pder f x + i\, \pder f y)\,\dzbar.
\end{align*}
Therefore, it is natural to introduce the following definitions of ``partial derivatives with respect to $z$ and $\zbar$'':
\begin{align*}
  \pder f z &= \half (\pder f x - i\, \pder f y)\\
  \pder f \zbar &= \half (\pder f x + i\, \pder f y)
\end{align*}
We can then write $\df = \pder f z \,\dz + \pder f \zbar \,\dzbar$.

Let us now express these in components, writing $f(x,y) = u(x,y) + i\,v(x,y)$ for real-valued functions $u$ and $v$.
Then we have
\begin{align*}
  \pder f x &= \pder u x + i\, \pder v x\\
  \pder f y &= \pder u y + i\, \pder v y
\end{align*}
and therefore
\begin{align*}
  \pder f z &= \half (\pder u x + \pder v y) + \half i\, (\pder v x - \pder u y)\\
  \pder f \zbar &= \half (\pder u x - \pder v y) + \half i\, (\pder v x + \pder u y)
\end{align*}

Now recall the \emph{Cauchy--Riemann equations} say that if $f$ is complex-differentiable (at a point), then
\[ \pder f y = i\,\pder f x \qquad\text{or equivalently}\qquad \pder f x = - i\,\pder f y,\]
(at that point) and conversely as long as $\pder f x$ and $\pder f y$ are continuous.
In components, this equation becomes
\[ \pder u x = \pder v y \qquad\text{and}\qquad \pder u y = - \pder v x. \]
In terms of the notations introduced above, these equations say exactly that
\[ \pder f \zbar = 0. \]
Thus, intuitively, for $f$ to be complex-differentiable means exactly that it is ``constant with respect to $\zbar$'' or ``a function of $z$ only''.

Finally, the equation $\pder f y = i\,\pder f x$ also implies that
\[ \pder f z = \half(\pder f x - i\,\pder f y) = \half (\pder f x + \pder f x) = \pder f x.\]
This is also the \emph{complex derivative} $f'(z)$ of $f$.
Thus, to say that $f$ is complex-differentiable is to say that we can write
\[ \df = f'(z) \,\dz \]
for some function $f'(z)$.
This is what we would expect to see for the ``differential'' of $f$ if we want to treat $z = x+i\,y$ as a \emph{single complex variable} rather than two real variables.


\section{Complex 1-forms}
\label{sec:complex-1-forms}

As shown above, a general complex-valued 1-form can be written as
\[f(x,y) \,\dz + g(x,y)\,\dzbar\]
for some complex-valued functions $f$ and $g$.
However, the above discussion of differentials, and the general philosophy of treating $z=x+i\,y$ as a single complex variable, suggest that we should really restrict our attention to 1-forms of the form
\[ f(z)\,\dz \]
for which $f$ is complex differentiable (in some region), i.e. $\pder f \zbar = 0$.
We call these \textbf{analytic 1-forms} (in that region).
They have the interesting property that \emph{their} differentials are always zero!
\begin{align*}
  \ed(f(z)\,\dz) &= \df \wedge \dz\\
  &= f'(z)\, \dz \wedge \dz\\
  &= 0
\end{align*}
As we will see, this is crucial to the amazing properties of complex-differentiable functions.

Note that there is no meaningful notion of ``analytic 2-form''.
This is in keeping with the idea that the complex plane has only ``one complex dimension'', even though it has two real dimensions.