Integrability in the complex case

We introduce the theory of integrable functions in ${\displaystyle \mathbb {C} }$. From the integrability of a function it will follow that it is differentiable infinite times, where it is holomorphic.

We will not consider generic integrals, but integrals along paths.

Therefore, we have to define ${\displaystyle curves}$ and ${\displaystyle paths}$ in ${\displaystyle \mathbb {C} }$.

Definition

A curve in ${\displaystyle \mathbb {C} }$ is a function ${\displaystyle \gamma :[a,b]\rightarrow \mathbb {C} }$ such that:

${\displaystyle \forall \,t\in [a,b],\;\gamma (t)\in \mathbb {C} .}$
Definition
The set ${\displaystyle \gamma ^{\star }=\gamma ([a,b])=\{\gamma (t)\in \mathbb {C} \colon t\in [a,b]\}}$ is edfined as the support of the curve ${\displaystyle \gamma }$. The points ${\displaystyle \gamma (a)}$ and ${\displaystyle \gamma (b)}$ are respectivelty the first and the second extremum of the curve. If ${\displaystyle \gamma (a)=\gamma (b)}$ we say that the curve is closed.''
Definition
A curve ${\displaystyle \gamma }$ in ${\displaystyle \mathbb {C} }$ is a path if it is piecewise differentiable.

Now, we can give a meaning to the expression integrate along a path. The definition we will give is the same given in the course of Multivariable Calculus for the elements of an oriented path. All the remarks given in the case of a multivariable function follow: among them, we have the invariance of the integral with respect to the chosen oriented path. In the following, we can take ${\displaystyle [a,b]\equiv [0,1]}$.

Definition

Let ${\displaystyle \gamma :[a,b]\rightarrow \mathbb {C} }$ be a path and ${\displaystyle f:{\mathcal {D}}\subset \mathbb {C} \rightarrow \mathbb {C} }$ a complex-valued function. We define integral of f along the path ${\displaystyle \gamma }$ the quantity:

${\displaystyle \int _{\gamma }f(z)dz=\int _{a}^{b}f(\gamma (t))\gamma ^{\prime }(t)dt.}$
Definition

We define the length of the curve ( or the path ) ${\displaystyle \gamma }$ as the quantity:

${\displaystyle L(\gamma )\equiv \int _{0}^{1}\mid \gamma ^{\prime }(t)\mid dt.}$

We have that:

${\displaystyle \mid \int _{\gamma }f(z)dz\mid \leq ML(\gamma ),\;{\mbox{where}}M\equiv max\{f(z)\colon z\in \gamma ^{\star }\ .}$

An important concept, that distinguishes the integral along paths in ${\displaystyle \mathbb {C} }$ from the one in ${\displaystyle \mathbb {R} ^{2}}$ is the index. In particular, we have the following:

Definition

Let ${\displaystyle \gamma :[a,b]\rightarrow \mathbb {C} }$ be a closed path and ${\displaystyle \Omega =\mathbb {C} \setminus \gamma ^{\star }}$. ${\displaystyle \forall \,z\in \Omega }$ we define index of z with respect to ${\displaystyle \gamma }$ the quantity:

${\displaystyle Ind_{\gamma }(z)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {d\xi }{\xi -z}}\equiv {\frac {1}{2\pi i}}\oint dt{\frac {\gamma ^{\prime }(t)}{\gamma (t)-z}}.}$

In this defintion we go through ${\displaystyle \gamma }$ in a counter-clockwise way. We set ${\displaystyle Ind_{\gamma }(\infty )=0.}$ We can see that every paths ${\displaystyle \gamma }$ in ${\displaystyle \mathbb {C} }$ satisfy an important property, analogous to the one given by the Jordan Theorem for simple and closed curves in ${\displaystyle \mathbb {R} ^{2}}$.In fact, we have that ${\displaystyle \gamma ^{\star }}$ always divides ${\displaystyle \mathbb {C} }$ in two connected components, one bounded and the other unbounded. There is an important relation between the function ${\displaystyle Ind_{\gamma }(z)}$ and the condition of holomorphism, as given by the following

Theorem
${\displaystyle Ind_{\gamma }(z)}$ is an holomorphic function in ${\displaystyle \mathbb {C} }$. Moreover, we have that ${\displaystyle Ind_{\gamma }\colon \Omega \subset \mathbb {C} \rightarrow \mathbb {Z} }$, that is ${\displaystyle Ind_{\gamma }}$ is an interger valued function and has constant valued on each connencted component in which ${\displaystyle \mathbb {C} }$ is splitted by ${\displaystyle \gamma }$. On the exterior of ${\displaystyle \gamma }$ the index is always zero.

As for the definition of interior and exterior, we follow the Jordan Theorem for simple and closed curves in ${\displaystyle \mathbb {R} ^{2}.}$ We can refer to books of Multivariate Calucus about this topic.

We provide a proof of the previous theorem in a simple case:

Proof:

Let ${\displaystyle \xi \in \mathbb {C} }$ and ${\displaystyle \gamma }$ be a circumference with centre ${\displaystyle z_{0}}$ and radius ${\displaystyle r}$. So, we have that ${\displaystyle \gamma (t)=re^{it}+z_{0},\,t\in [0,2\pi ].}$ We obtain that:

${\displaystyle Ind_{\gamma }(z)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {d\xi }{\xi -z}}={\frac {1}{2\pi i}}\int _{0}^{2\pi }{\frac {\gamma ^{\prime }(t)}{\gamma (t)-z}}dt={\frac {1}{2\pi i}}\int _{0}^{2\pi }{\frac {ire^{it}}{re^{it}}}dt=1.}$

${\displaystyle \Box }$

We remind the defintions of connected, convex & simply connected set.

Definition
We say that ${\displaystyle \Omega \subset \mathbb {C} }$ is connected if:
${\displaystyle \forall \,z_{1},z_{2}\in \Omega }$ we have that ${\displaystyle z(t)=(1-t)z_{1}+tz_{2}\subset \Omega .}$

In general, for the other twoue definitions, we refer to textbooks of Multivariate Calculus, where one can find rigorous definitions for ${\displaystyle \Omega \subset \mathbb {R} ^{n},\,n\geq 2}$ that can be easily extended to ${\displaystyle \mathbb {C} .}$

We state two of the most important results in complex analysis, given the notion:

${\displaystyle {\hat {\mathbb {C} }}\equiv \mathbb {C} \cup \{\infty \}.}$

Theorem

We consider ${\displaystyle \Omega \subset {\hat {\mathbb {C} }}}$ and ${\displaystyle f}$ an holomorphic function in ${\displaystyle \Omega .}$ If ${\displaystyle \gamma }$ is a closed path in ${\displaystyle \Omega }$ such that ${\displaystyle Ind_{\gamma }(z)=0,\,\forall \,z\in {\hat {\mathbb {C} }}\setminus \Omega ,}$ then:

${\displaystyle \int _{\gamma }f(z)dz=0.}$

Equivalently, we have that:

Let ${\displaystyle \Omega \subset {\hat {\mathbb {C} }}}$ be a simply connected set (and so ${\displaystyle {\hat {\mathbb {C} }}\setminus \Omega }$ is connected) and ${\displaystyle f}$ be a holomorphic function in ${\displaystyle \Omega .}$ If ${\displaystyle \gamma }$ is a closed path in ${\displaystyle \Omega }$, then:

${\displaystyle \int _{\gamma }f(z)dz=0.}$

We see that the theorem provides a necessary condition for the integral along the path to be zero, in fact we could have cases in which the curve has a non-zero index and so we could not say anything about the value of the integral a priori.

An important corollary of Cauchy theorem gives conditions for the comparison between integrals of the same holomorphic ${\displaystyle f}$ along two different closed paths in ${\displaystyle \Omega }$.

Theorem
We consider ${\displaystyle \Omega \subset {\hat {\mathbb {C} }}}$ and ${\displaystyle f}$ an holomorphic function in ${\displaystyle \Omega .}$ Let ${\displaystyle \gamma _{1}\;\&\;\gamma _{2}}$ be two closed paths in ${\displaystyle \Omega }$ such that ${\displaystyle Ind_{\gamma _{1}}(z)=Ind_{\gamma _{2}}(z),\;\forall \,z\in {\hat {\mathbb {C} }}\setminus \Omega }$, then:
${\displaystyle \int _{\gamma _{1}}f(z)dz=\int _{\gamma _{2}}f(z)dz.}$

Proof:

Let ${\displaystyle f}$ be holomorphic in ${\displaystyle \Omega }$and ${\displaystyle \gamma ,\lambda }$ be two closed paths. We link ${\displaystyle \gamma }$ and ${\displaystyle \lambda }$ with two line segments ${\displaystyle \pm c}$ with a distance ${\displaystyle \epsilon }$ between them. It is not possible to have a closed path with zero index with respect to every ${\displaystyle z\in {\hat {\mathbb {C} }}\setminus \Omega }$. Let such ${\displaystyle \Gamma }$ given by: ${\displaystyle \Gamma =\gamma \,\cup \,c\,\cup \,-\lambda \,\cup \,-c.}$ By applying Cauchy's theorem to the function ${\displaystyle f}$ considered:

${\displaystyle 0=\int _{\Gamma }f(z)dz=\int _{\gamma }f(z)dz+\int _{c}f(z)dz+\int _{(-\lambda )}f(z)dz+\int _{(-c)}f(z)dz=}$
${\displaystyle =\int _{\gamma }f(z)dz+\int _{c}f(z)dz-\int _{\lambda }f(z)dz-\int _{c}f(z)dz=}$
${\displaystyle =\int _{\gamma }f(z)dz-\int _{\lambda }f(z)dz.}$
It follows that: ${\displaystyle \int _{\gamma }f(z)dz=\int _{\lambda }f(z)dz.}$

${\displaystyle \Box }$