Differential forms / 05

Complex line integrals

The forms $$dz$$ and $$d\bar z$$ are a change of basis for planar 1-forms; residues are period computations around punctures.

Calculus root: complex notation for two real line integrals

A complex line integral is still a line integral. If $z=x+iy$ and $f=u+iv$, then

\[f(z)\,dz=(u\,dx-v\,dy)+i(v\,dx+u\,dy).\]

Thus one complex integral packages two real 1-form integrals. The new power of the notation is that holomorphicity imposes the Cauchy-Riemann equations, which turn those two real forms into closed forms.

Example: the integral of $z\,dz$

Since

\[z\,dz=d\left({z^2\over2}\right),\]

its integral around every closed contour is zero. This is the complex version of integrating an exact 1-form.

Example: $dz/z$ and the missing primitive

On a small simply connected sector of $\mathbb C^*$,

\[{dz\over z}=d\log z.\]

Around the unit circle, however,

\[\int_{|z|=1}{dz\over z}=2\pi i.\]

The local primitive exists, but it does not glue globally. This is the same obstruction that later appears in sheaf cohomology through local logarithms.

Section 4.3 of the PDF rewrites planar line integrals in complex coordinates. Let

\[z=x+iy,\qquad \bar z=x-iy.\]

Then

\[dz=dx+i\,dy,\qquad d\bar z=dx-i\,dy,\]

and conversely

\[dx={1\over 2}(dz+d\bar z),\qquad dy={1\over 2i}(dz-d\bar z).\]

Real 1-forms in complex basis

For a real 1-form $\eta=P\,dx+Q\,dy$,

\[\eta={1\over 2}(P-iQ)\,dz+{1\over 2}(P+iQ)\,d\bar z.\]

This is linear algebra, not yet holomorphic analysis. Holomorphicity enters when the coefficient of $d\bar z$ disappears in the appropriate differential equation.

Example 1: convert a real form

For $\eta=x\,dx-y\,dy$, use the formulas above:

\[\eta={1\over 2}\bigl((x+iy)\,dz+(x-iy)\,d\bar z\bigr) ={1\over 2}(z\,dz+\bar z\,d\bar z).\]

Since $\eta=d\left({1\over2}(x^2-y^2)\right)$, it is exact.

Example 2: the form $d\bar z\wedge dz$

Compute

\[d\bar z\wedge dz=(dx-i\,dy)\wedge(dx+i\,dy)=2i\,dx\wedge dy.\]

Thus

\[dx\wedge dy={1\over 2i}d\bar z\wedge dz.\]

This identity is the area-form bridge used in the Cauchy-Green formula.

The vortex form as an imaginary part

On $\mathbb C^\times$,

\[{dz\over z} ={dx+i\,dy\over x+iy} ={x\,dx+y\,dy\over x^2+y^2} +i\,{x\,dy-y\,dx\over x^2+y^2}.\]

Therefore

\[\operatorname{Im}{dz\over z} ={x\,dy-y\,dx\over x^2+y^2}.\]

This is exactly the closed non-exact form from the punctured-plane example.

For $C_R(t)=Re^{it}$,

\[{dz\over z}=i\,dt, \qquad \int_{C_R}{dz\over z}=2\pi i, \qquad \int_{C_R}\operatorname{Im}{dz\over z}=2\pi.\]

Winding number

For a loop $C$ avoiding $p$,

\[{1\over 2\pi i}\int_C {dz\over z-p}\]

is the winding number of $C$ about $p$ when $C$ is a standard piecewise smooth closed curve. In form language, the integral of $dz/(z-p)$ is a period of a closed form on $\mathbb C\setminus\{p\}$.

Example 3: an $n$-fold loop

Let $C_n(t)=p+Re^{int}$, $0\le t\le 2\pi$. Then

\[dz=i n Re^{int}\,dt,\qquad z-p=Re^{int},\]

$\int_{C_n}{dz\over z-p}=2\pi i n.$

Example 4: residue as a coefficient

For

\[g(z)={3\over z-a}+{2\over (z-a)^2}+h(z)\]

with $h$ holomorphic near $a$, the integral around a small positively oriented circle is

\[\int g(z)\,dz=2\pi i\cdot 3.\]

The double-pole term contributes no period because it is the derivative of $-2/(z-a)$.

Relation to Cauchy’s theorem

If $g$ is holomorphic on a simply connected domain, then $g(z)\,dz$ has a global primitive and all closed contour integrals vanish. If the domain has isolated poles, the obstruction is measured by residues and winding numbers:

$\int_C g(z)\,dz=2\pi i\sum_a \operatorname{Res}_a(g)\operatorname{wind}(C,a),$ where the sum is over poles $a$ not on $C$, with only poles of nonzero winding contributing.

This is not separate from differential forms. It is the same closed-versus-exact story in complex notation.