Integrability in the complex case

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Therefore, we have to define <math>curves</math> and <math>paths</math> in <math>\mathbb {C}</math>.
 
Therefore, we have to define <math>curves</math> and <math>paths</math> in <math>\mathbb {C}</math>.
  
<definition>A ''curve'' in <math>\mathbb {C}</math> is a function <math>\gamma :[a,b]\rightarrow \mathbb {C}</math> such that:
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{{Definition|A ''curve'' in <math>\mathbb {C}</math> is a function <math>\gamma :[a,b]\rightarrow \mathbb {C}</math> such that:
  
<math display="block">  \forall \, t\in [a,b],\; \gamma (t)\in \mathbb {C}.  </math> </definition>
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<math display="block">  \forall \, t\in [a,b],\; \gamma (t)\in \mathbb {C}.  </math>
<definition>The set <math>\gamma ^{\star }=\gamma ([a,b])=\{ \gamma (t)\in \mathbb {C}\colon t\in [a,b]\} </math> is edfined as the ''support of the curve'' <math>\gamma </math>. The points <math>\gamma (a)</math> and <math>\gamma (b)</math> are respectivelty the first and the second extremum of the curve. If <math>\gamma (a)=\gamma (b)</math> we say that the curve is ''closed.''</definition>
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}}
  
<definition> A curve  <math>\gamma </math> in <math>\mathbb {C}</math> is a ''path'' if it is piecewise differentiable.</definition>
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{{Definition|
 +
The set <math>\gamma ^{\star }=\gamma ([a,b])=\{ \gamma (t)\in \mathbb {C}\colon t\in [a,b]\} </math> is edfined as the ''support of the curve'' <math>\gamma </math>. The points <math>\gamma (a)</math> and <math>\gamma (b)</math> are respectivelty the first and the second extremum of the curve. If <math>\gamma (a)=\gamma (b)</math> we say that the curve is ''closed.''
 +
}}
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 +
{{Definition| A curve  <math>\gamma </math> in <math>\mathbb {C}</math> 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 <math>[a,b]\equiv [0,1]</math>.
 
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 <math>[a,b]\equiv [0,1]</math>.
  
<definition>Let <math>\gamma :[a,b]\rightarrow \mathbb {C}</math> be a path and <math>f:\mathcal{D}\subset \mathbb {C}\rightarrow \mathbb {C}</math> a complex-valued function. We define ''integral of f along the path'' <math>\gamma </math> the quantity:
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{{Definition|Let <math>\gamma :[a,b]\rightarrow \mathbb {C}</math> be a path and <math>f:\mathcal{D}\subset \mathbb {C}\rightarrow \mathbb {C}</math> a complex-valued function. We define ''integral of f along the path'' <math>\gamma </math> the quantity:
  
<math display="block">  \int _{\gamma }f(z)dz=\int _{a}^{b}f(\gamma (t))\gamma ^{\prime }(t)dt.  </math></definition>
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<math display="block">  \int _{\gamma }f(z)dz=\int _{a}^{b}f(\gamma (t))\gamma ^{\prime }(t)dt.  </math>
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}}
  
<definition>We define the length of the curve ( or the path ) <math>\gamma </math> as the quantity:
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{{Definition|
 +
We define the length of the curve ( or the path ) <math>\gamma </math> as the quantity:
  
<math display="block">  L(\gamma )\equiv \int _{0}^{1}\mid \gamma ^{\prime }(t)\mid dt.  </math></definition>
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<math display="block">  L(\gamma )\equiv \int _{0}^{1}\mid \gamma ^{\prime }(t)\mid dt.  </math>
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}}
  
 
We have that:
 
We have that:
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An important concept, that distinguishes the integral along paths in <math>\mathbb {C}</math> from the one in <math>\mathbb {R}^{2}</math> is the ''index.'' In particular, we have the following:
 
An important concept, that distinguishes the integral along paths in <math>\mathbb {C}</math> from the one in <math>\mathbb {R}^{2}</math> is the ''index.'' In particular, we have the following:
  
<definition>Let <math>\gamma :[a,b]\rightarrow \mathbb {C}</math> be a closed path and <math>\Omega =\mathbb {C}\setminus \gamma ^{\star }</math>. <math>\forall \, z\in \Omega </math> we define ''index of z with respect to'' <math>\gamma </math> the quantity:
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{{Definition|
 +
Let <math>\gamma :[a,b]\rightarrow \mathbb {C}</math> be a closed path and <math>\Omega =\mathbb {C}\setminus \gamma ^{\star }</math>. <math>\forall \, z\in \Omega </math> we define ''index of z with respect to'' <math>\gamma </math> the quantity:
  
<math display="block">  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}.  </math></definition>
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<math display="block">  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}.  </math>
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}}
  
 
In this defintion we go through <math>\gamma </math> in a counter-clockwise way. We set <math>Ind_{\gamma }(\infty )=0.</math>
 
In this defintion we go through <math>\gamma </math> in a counter-clockwise way. We set <math>Ind_{\gamma }(\infty )=0.</math>
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There is an important relation between the function <math>Ind_{\gamma }(z)</math> and the condition of holomorphism, as given by the following
 
There is an important relation between the function <math>Ind_{\gamma }(z)</math> and the condition of holomorphism, as given by the following
  
<theorem><math>Ind_{\gamma }(z)</math> is an holomorphic function in <math>\mathbb {C}</math>. Moreover, we have that <math>Ind_{\gamma }\colon \Omega \subset \mathbb {C}\rightarrow \mathbb {Z}</math>, that is <math>Ind_{\gamma }</math> is an interger valued function and has constant valued on each connencted component in which <math>\mathbb {C}</math> is splitted by <math>\gamma </math>. On the exterior of <math>\gamma </math> the index is always zero.</theorem>
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{{Theorem|
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<math>Ind_{\gamma }(z)</math> is an holomorphic function in <math>\mathbb {C}</math>. Moreover, we have that <math>Ind_{\gamma }\colon \Omega \subset \mathbb {C}\rightarrow \mathbb {Z}</math>, that is <math>Ind_{\gamma }</math> is an interger valued function and has constant valued on each connencted component in which <math>\mathbb {C}</math> is splitted by <math>\gamma </math>. On the exterior of <math>\gamma </math> the index is always zero.
 +
}}
  
 
As for the definition of interior and exterior, we follow the ''Jordan Theorem'' for simple and closed curves in <math>\mathbb {R}^{2}.</math> We can refer to books of Multivariate Calucus about this topic.
 
As for the definition of interior and exterior, we follow the ''Jordan Theorem'' for simple and closed curves in <math>\mathbb {R}^{2}.</math> We can refer to books of Multivariate Calucus about this topic.
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We provide a proof of the previous theorem in a simple case:
 
We provide a proof of the previous theorem in a simple case:
  
<proof>Let <math>\xi \in \mathbb {C}</math> and <math>\gamma </math> be a circumference with centre <math>z_{0}</math> and radius <math>r</math>. So, we have that <math>\gamma (t)=re^{it}+z_{0},\, t\in [0,2\pi ].</math>
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{{Proof|Let <math>\xi \in \mathbb {C}</math> and <math>\gamma </math> be a circumference with centre <math>z_{0}</math> and radius <math>r</math>. So, we have that <math>\gamma (t)=re^{it}+z_{0},\, t\in [0,2\pi ].</math>
  
 
We obtain that:
 
We obtain that:
  
 
<math display="block">  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.  </math>
 
<math display="block">  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.  </math>
</proof>
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}}
  
 
We remind the defintions of ''connected, convex & simply connected'' set.
 
We remind the defintions of ''connected, convex & simply connected'' set.
  
<definition>We say that <math>\Omega \subset \mathbb {C}</math> is connected if:<br/> <math>\forall \, z_{1},z_{2}\in \Omega </math> we have that <math>z(t)=(1-t)z_{1}+tz_{2}\subset \Omega .</math></definition>
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{{Definition|We say that <math>\Omega \subset \mathbb {C}</math> is connected if:<br/> <math>\forall \, z_{1},z_{2}\in \Omega </math> we have that <math>z(t)=(1-t)z_{1}+tz_{2}\subset \Omega .</math>
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}}
  
 
In general, for the other twoue definitions, we refer to textbooks of Multivariate Calculus, where one can find rigorous definitions for <math>\Omega \subset \mathbb {R}^{n},\, n\geq 2</math> that can be easily extended to <math>\mathbb {C}.</math>
 
In general, for the other twoue definitions, we refer to textbooks of Multivariate Calculus, where one can find rigorous definitions for <math>\Omega \subset \mathbb {R}^{n},\, n\geq 2</math> that can be easily extended to <math>\mathbb {C}.</math>
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<math display="block">  \hat{\mathbb {C}}\equiv \mathbb {C}\cup \{ \infty \} .  </math>
 
<math display="block">  \hat{\mathbb {C}}\equiv \mathbb {C}\cup \{ \infty \} .  </math>
  
<theorem> We consider <math>\Omega \subset \hat{\mathbb {C}}</math> and <math>f</math> an holomorphic function in <math>\Omega .</math> If <math>\gamma </math> is a closed path in <math>\Omega </math> such that <math>Ind_{\gamma }(z)=0,\, \forall \, z\in \hat{\mathbb {C}}\setminus \Omega ,</math> then:
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{{Theorem| We consider <math>\Omega \subset \hat{\mathbb {C}}</math> and <math>f</math> an holomorphic function in <math>\Omega .</math> If <math>\gamma </math> is a closed path in <math>\Omega </math> such that <math>Ind_{\gamma }(z)=0,\, \forall \, z\in \hat{\mathbb {C}}\setminus \Omega ,</math> then:
  
 
<math display="block">  \int _{\gamma }f(z)dz=0.  </math>
 
<math display="block">  \int _{\gamma }f(z)dz=0.  </math>
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Let <math>\Omega \subset \hat{\mathbb {C}}</math> be a ''simply connected'' set (and so <math>\hat{\mathbb {C}}\setminus \Omega </math> is connected) and <math>f</math> be a holomorphic function in <math>\Omega .</math> If <math>\gamma </math> is a closed path in <math>\Omega </math>, then:
 
Let <math>\Omega \subset \hat{\mathbb {C}}</math> be a ''simply connected'' set (and so <math>\hat{\mathbb {C}}\setminus \Omega </math> is connected) and <math>f</math> be a holomorphic function in <math>\Omega .</math> If <math>\gamma </math> is a closed path in <math>\Omega </math>, then:
  
<math display="block">  \int _{\gamma }f(z)dz=0.  </math> </theorem>
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<math display="block">  \int _{\gamma }f(z)dz=0.  </math>  
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}}
  
 
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.
 
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.
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An important corollary of Cauchy theorem gives conditions for the comparison between integrals of the same holomorphic <math>f</math> along two different closed paths in <math>\Omega </math>.
 
An important corollary of Cauchy theorem gives conditions for the comparison between integrals of the same holomorphic <math>f</math> along two different closed paths in <math>\Omega </math>.
  
<theorem>We consider <math>\Omega \subset \hat{\mathbb {C}}</math> and <math>f</math> an holomorphic function in <math>\Omega .</math> Let <math>\gamma _{1}\; \& \; \gamma _{2}</math> be two closed paths in <math>\Omega </math> such that <math>Ind_{\gamma _{1}}(z)=Ind_{\gamma _{2}}(z),\; \forall \, z\in \hat{\mathbb {C}}\setminus \Omega </math>, then: <math display="block">  \int _{\gamma _{1}}f(z)dz=\int _{\gamma _{2}}f(z)dz.  </math></theorem>
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{{Theorem|We consider <math>\Omega \subset \hat{\mathbb {C}}</math> and <math>f</math> an holomorphic function in <math>\Omega .</math> Let <math>\gamma _{1}\; \& \; \gamma _{2}</math> be two closed paths in <math>\Omega </math> such that <math>Ind_{\gamma _{1}}(z)=Ind_{\gamma _{2}}(z),\; \forall \, z\in \hat{\mathbb {C}}\setminus \Omega </math>, then: <math display="block">  \int _{\gamma _{1}}f(z)dz=\int _{\gamma _{2}}f(z)dz.  </math>
 +
}}
  
<proof> Let <math>f</math> be holomorphic in <math>\Omega </math>and <math>\gamma ,\lambda </math> be two closed paths.
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{{Proof| Let <math>f</math> be holomorphic in <math>\Omega </math>and <math>\gamma ,\lambda </math> be two closed paths.
  
 
We link <math>\gamma </math> and <math>\lambda </math> with two line segments <math>\pm c</math> with a distance <math>\epsilon </math> between them.
 
We link <math>\gamma </math> and <math>\lambda </math> with two line segments <math>\pm c</math> with a distance <math>\epsilon </math> between them.
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It follows that: <math>\int _{\gamma }f(z)dz=\int _{\lambda }f(z)dz.</math>
 
It follows that: <math>\int _{\gamma }f(z)dz=\int _{\lambda }f(z)dz.</math>
</proof>
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}}

Revision as of 08:51, 21 October 2016

We introduce the theory of integrable functions in . 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 and in .

Definition

A curve in is a function such that:

Definition
The set is edfined as the support of the curve . The points and are respectivelty the first and the second extremum of the curve. If we say that the curve is closed.''
Definition
A curve in 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 .

Definition

Let be a path and a complex-valued function. We define integral of f along the path the quantity:

Definition

We define the length of the curve ( or the path ) as the quantity:

We have that:

An important concept, that distinguishes the integral along paths in from the one in is the index. In particular, we have the following:

Definition

Let be a closed path and . we define index of z with respect to the quantity:

In this defintion we go through in a counter-clockwise way. We set We can see that every paths in satisfy an important property, analogous to the one given by the Jordan Theorem for simple and closed curves in .In fact, we have that always divides in two connected components, one bounded and the other unbounded. There is an important relation between the function and the condition of holomorphism, as given by the following


Theorem
is an holomorphic function in . Moreover, we have that , that is is an interger valued function and has constant valued on each connencted component in which is splitted by . On the exterior of the index is always zero.


As for the definition of interior and exterior, we follow the Jordan Theorem for simple and closed curves in 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 and be a circumference with centre and radius . So, we have that We obtain that:

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

Definition
We say that is connected if:
we have that

In general, for the other twoue definitions, we refer to textbooks of Multivariate Calculus, where one can find rigorous definitions for that can be easily extended to

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


Theorem

We consider and an holomorphic function in If is a closed path in such that then:

Equivalently, we have that:

Let be a simply connected set (and so is connected) and be a holomorphic function in If is a closed path in , then:


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 along two different closed paths in .


Theorem
We consider and an holomorphic function in Let be two closed paths in such that , then:


Proof:

Let be holomorphic in and be two closed paths. We link and with two line segments with a distance between them. It is not possible to have a closed path with zero index with respect to every . Let such given by: By applying Cauchy's theorem to the function considered:

It follows that:

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