Cauchy's integral theorem

The Cauchy's integral theorem (after Augustin Louis Cauchy ) is one of the key sets of function theory . It deals with curve integrals for holomorphic (complex-differentiable on an open set) functions. In essence, it says that two paths connecting the same points have the same path integral if the function is holomorphic everywhere between the two paths. The theorem derives its meaning from the fact that it is used to prove Cauchy's integral formula and the residual theorem .

The first formulation of the theorem dates from 1814 , when Cauchy proved it for rectangular areas. He generalized this over the next few years, although he took the Jordanian curve set for granted. Thanks to Goursat's lemma, modern proofs get by without this profound statement from the topology .

The sentence

The integral theorem was formulated in numerous versions.

Cauchy's integral theorem for elementary domains

Let be an elementary domain, i.e. a domain in which every holomorphic function has an antiderivative . Star regions are, for example, elementary regions . Cauchy's integral theorem now says that ${\displaystyle D\subseteq \mathbb {C} }$ ${\displaystyle f\colon D\to \mathbb {C} }$

${\displaystyle \oint _{\gamma }f(z)\,dz=0}$

for each closed curve (where and ). For the integral sign with circle see notation for curve integrals of closed curves . ${\displaystyle \gamma \colon [a,b]\to D}$ ${\displaystyle a,b\in \mathbb {R} }$ ${\displaystyle a<b}$

If there is no elementary area, the statement is false. For example, the field is holomorphic, but it does not vanish over every closed curve. For example ${\displaystyle D}$ ${\displaystyle f\colon z\mapsto {\tfrac {1}{z}}}$ ${\displaystyle \mathbb {C} \setminus \{0\}}$ ${\displaystyle \textstyle \oint _{\gamma }f(z)\,dz}$

${\displaystyle \oint _{\partial U_{r}(0)}{\frac {1}{z}}\mathrm {d} z=2\pi \mathrm {i} \neq 0}$

for the simply traversed edge curve of a circular disk with a positive radius . ${\displaystyle 0}$ ${\displaystyle r}$

Cauchy's integral theorem (homotopy version)

Is open and two homotopic curves are in , then is ${\displaystyle D\subseteq \mathbb {C} }$ ${\displaystyle \alpha ,\beta \colon [0,1]\to D}$ ${\displaystyle D}$

${\displaystyle \int \limits _{\alpha }f(z)\,dz=\int \limits _{\beta }f(z)\,dz}$

for every holomorphic function . ${\displaystyle f\colon D\to \mathbb {C} }$

Is a simply connected region, then the integral vanishes after homotopies version for each closed curve d. H. is an elementary area . ${\displaystyle D}$ ${\displaystyle D}$

If you look back at the example above, you notice that is not simply connected. ${\displaystyle \mathbb {C} \setminus \{0\}}$

Cauchy's integral theorem (homology version)

If an area and a cycle is in , then it disappears ${\displaystyle D\subseteq \mathbb {C} }$ ${\displaystyle \Gamma }$ ${\displaystyle D}$

${\displaystyle \int \limits _{\Gamma }f(z)\,dz}$

for every holomorphic function if and only if is zero homolog in . ${\displaystyle f\colon D\to \mathbb {C} }$ ${\displaystyle \Gamma }$ ${\displaystyle D}$

Isolated singularities

Number of turns of the integration path

Let it be an area, an interior point, and holomorphic. Let be a dotted neighborhood that is holomorphic. Furthermore, let it be a completely closed curve that revolves exactly once with a positive orientation, i.e. H. for the circulation rate applies (in particular is not on ). With the integral theorem, we now have ${\displaystyle D\subseteq \mathbb {C} }$ ${\displaystyle a\in D}$ ${\displaystyle f\colon D\setminus \{a\}\to \mathbb {C} }$ ${\displaystyle U:=U_{r}(a)\setminus \{a\}\subset D}$ ${\displaystyle f}$ ${\displaystyle \gamma }$ ${\displaystyle D}$ ${\displaystyle a}$ ${\displaystyle \operatorname {ind} _{\gamma }(a)=1}$ ${\displaystyle a}$ ${\displaystyle \gamma }$

${\displaystyle \oint _{\gamma }f(z)\,dz=\oint _{\partial U}f(z)\,dz.}$

By generalizing to any number of circulation , one obtains ${\displaystyle \gamma }$

${\displaystyle \oint _{\gamma }f(z)\,dz=\operatorname {ind} _{\gamma }(a)\oint _{\partial U}f(z)\,dz.}$

With the help of the definition of the residual it even results

${\displaystyle {\frac {1}{2\pi \mathrm {i} }}\oint _{\gamma }f(z)\,dz=\operatorname {ind} _{\gamma }(a)\operatorname {Res} _{a}f(z).}$

The residual theorem is a generalization of this approach to several isolated singularities and to cycles.

example

The integral is below with determined. As integration path Choose a circle with radius to , so ${\displaystyle \oint _{\partial U(a)}{\frac {1}{(z-a)^{n}}}\mathrm {d} z}$ ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle \partial U(a)=\partial U_{r}(a)}$ ${\displaystyle r}$ ${\displaystyle a}$

${\displaystyle z=\gamma (t)=a+re^{2\pi \mathrm {i} t}\quad \Rightarrow \quad \mathrm {d} z={\frac {\partial \gamma }{\partial t}}\mathrm {d} t=2\pi ire^{2\pi \mathrm {i} t}\mathrm {d} t}$

When used, results in:

${\displaystyle {\begin{aligned}\oint _{\partial U_{r}(a)}{\frac {1}{(z-a)^{n}}}\mathrm {d} z&=\int _{0}^{1}{\frac {2\pi \mathrm {i} re^{2\pi \mathrm {i} t}}{r^{n}e^{2\pi n\mathrm {i} t}}}\mathrm {d} t=2\pi \mathrm {i} r^{1-n}\int _{0}^{1}e^{2\pi \mathrm {i} t(1-n)}\mathrm {d} t={\begin{cases}2\pi \mathrm {i} [t]_{0}^{1}&{\mbox{für}}\ n=1\\{\frac {r^{1-n}}{1-n}}[e^{2\pi \mathrm {i} t(1-n)}]_{0}^{1}&{\mbox{für}}\ n\neq 1\end{cases}}\\&={\begin{cases}2\pi \mathrm {i} &{\mbox{für}}\ n=1\\0&{\mbox{für}}\ n\neq 1\end{cases}}=2\pi \mathrm {i} \delta _{n,1}\end{aligned}}}$

Since every function that is holomorphic on a circular ring around can be expanded into a Laurent series , the integration results around : ${\displaystyle f(z)}$ ${\displaystyle a}$ ${\displaystyle f(z)=\sum _{n=-\infty }^{\infty }c_{n}(z-a)^{n}}$ ${\displaystyle a}$

${\displaystyle \oint _{\partial U(a)}f(z)\mathrm {d} z=\oint _{\partial U(a)}\sum _{n=-\infty }^{\infty }c_{n}(z-a)^{n}\mathrm {d} z=\sum _{n=-\infty }^{\infty }c_{n}\oint _{\partial U(a)}(z-a)^{n}\mathrm {d} z}$

The above result can now be used: ${\displaystyle \oint _{\partial U_{r}(a)}(z-a)^{n}\mathrm {d} z=2\pi \mathrm {i} \delta _{n,-1}}$

${\displaystyle \oint _{\partial U(a)}f(z)\mathrm {d} z=\sum _{n=-\infty }^{\infty }c_{n}2\pi \mathrm {i} \delta _{n,-1}=2\pi \mathrm {i} \,c_{-1}=2\pi \mathrm {i} \,{\text{Res}}_{a}(f)}$,

where the expansion coefficient was called the residual . ${\displaystyle c_{-1}}$

Derivation

The following derivation, which presupposes the continuous complex differentiability, leads the complex integral back to real two-dimensional integrals.

Be with and with . Then applies to the integral along the curve in the complex plane, or to the equivalent line integral along the curve ${\displaystyle z=x+iy\in \mathbb {C} }$ ${\displaystyle x,y\in \mathbb {R} }$ ${\displaystyle f(z)=f(x,y)=u(x,y)+iv(x,y)\in \mathbb {C} }$ ${\displaystyle u,v\in \mathbb {R} }$ ${\displaystyle \gamma (z)}$

${\displaystyle C(x,y)={\begin{pmatrix}\Re (\gamma (z))\\\Im (\gamma (z))\end{pmatrix}}={\begin{pmatrix}\Re (\gamma (x,y))\\\Im (\gamma (x,y))\end{pmatrix}}}$

in the real plane ${\displaystyle \mathbb {R} ^{2}}$

${\displaystyle {\begin{aligned}{\underset {\gamma \subset \mathbb {C} }{\int }}f(z)\,dz&={\underset {C\subset \mathbb {R} ^{2}}{\int }}f(x,y)\,(dx+idy)={\underset {C\subset \mathbb {R} ^{2}}{\int }}{\begin{pmatrix}f(x,y)\\if(x,y)\end{pmatrix}}\cdot {\begin{pmatrix}dx\\dy\end{pmatrix}}\\&={\underset {C\subset \mathbb {R} ^{2}}{\int }}{\begin{pmatrix}u(x,y)\\-v(x,y)\end{pmatrix}}\cdot {\begin{pmatrix}dx\\dy\end{pmatrix}}+i{\underset {C\subset \mathbb {R} ^{2}}{\int }}{\begin{pmatrix}v(x,y)\\u(x,y)\end{pmatrix}}\cdot {\begin{pmatrix}dx\\dy\end{pmatrix}}\end{aligned}}}$

The complex curve integral was thus expressed by two real curve integrals.

For a closed curve that borders a simply connected area S, Gauss's theorem (here the continuity of the partial derivatives is used) can be applied ${\displaystyle C=\partial S}$

${\displaystyle {\begin{aligned}{\underset {\gamma \subset \mathbb {C} }{\oint }}f(z)\,dz&={\underset {S\subset \mathbb {R} ^{2}}{\int }}{\begin{pmatrix}\partial _{x}\\\partial _{y}\end{pmatrix}}\cdot {\begin{pmatrix}u\\-v\end{pmatrix}}dxdy+i{\underset {S\subset \mathbb {R} ^{2}}{\int }}{\begin{pmatrix}\partial _{x}\\\partial _{y}\end{pmatrix}}\cdot {\begin{pmatrix}v\\u\end{pmatrix}}dxdy\\&={\underset {S\subset \mathbb {R} ^{2}}{\int }}\left\{\partial _{x}u-\partial _{y}v\right\}dxdy+i{\underset {S\subset \mathbb {R} ^{2}}{\int }}\left\{\partial _{x}v+\partial _{y}u\right\}dxdy\end{aligned}}}$

or alternatively the Stokes theorem

${\displaystyle {\begin{aligned}{\underset {\gamma \subset \mathbb {C} }{\oint }}f(z)\,dz&={\underset {S\subset \mathbb {R} ^{2}}{\int }}\left[{\begin{pmatrix}\partial _{x}\\\partial _{y}\\0\end{pmatrix}}\times {\begin{pmatrix}u\\-v\\0\end{pmatrix}}\right]_{3}dxdy+i{\underset {S\subset \mathbb {R} ^{2}}{\int }}\left[{\begin{pmatrix}\partial _{x}\\\partial _{y}\\0\end{pmatrix}}\times {\begin{pmatrix}v\\u\\0\end{pmatrix}}\right]_{3}dxdy\\&={\underset {S\subset \mathbb {R} ^{2}}{\int }}\left\{-\partial _{x}v-\partial _{y}u\right\}dxdy+i{\underset {S\subset \mathbb {R} ^{2}}{\int }}\left\{\partial _{x}u-\partial _{y}v\right\}dxdy\end{aligned}}}$

If the function in S is complex differentiable , the Cauchy-Riemann differential equations must there ${\displaystyle f(z)}$

${\displaystyle \partial _{x}u=\partial _{y}v}$   and   ${\displaystyle \partial _{x}v=-\partial _{y}u}$

hold, so that the integrands above (regardless of whether in the Gauss or Stokes version) vanish:

${\displaystyle {\underset {\gamma }{\oint }}f(z)\,dz=0}$

Thus, Cauchy's integral theorem for holomorphic functions on simply connected domains is proven.

Cauchy's integral theorem with Wirtinger calculus and Stokes' theorem

The Cauchy integral theorem is obtained as slight consequence of the set of Stokes when the Wirtinger derivatives can bring to bear. To prove the integral theorem, the calculation of the curve integral is understood as the integration of the complex-valued differential form

${\displaystyle \omega =f(z)dz}$

via the closed curve that runs around the simply contiguous and bordered area . ${\displaystyle C}$ ${\displaystyle C=\partial S}$ ${\displaystyle S}$

The Wirtinger calculus says that the differential is the representation ${\displaystyle df}$

${\displaystyle df={\frac {\partial f}{\partial z}}dz+{\frac {\partial f}{\partial {\bar {z}}}}{d{\bar {z}}}}$

has what immediately

${\displaystyle d{\omega }=df\wedge dz={{\frac {\partial f}{\partial z}}dz}\wedge dz+{{\frac {\partial f}{\partial {\bar {z}}}}{d{\bar {z}}}}\wedge dz}$

follows.

Well first is fundamental

${\displaystyle dz\wedge dz=0}$

Furthermore, the assumed means Holomorphiebedingung for after Wirtinger derivatives nothing more than ${\displaystyle f}$

${\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}=0}$  ,

what immediately

${\displaystyle {{\frac {\partial f}{\partial {\bar {z}}}}{d{\bar {z}}}}\wedge dz=0}$

entails.

So the overall result is:

${\displaystyle d{\omega }=0}$

and finally by means of Stokes' theorem :

${\displaystyle \int _{C}f(z)dz=\int _{\partial S}\omega =\int _{S}\mathrm {d} \omega =\int _{S}\mathrm {0} =0}$

annotation

With the help of Goursat's integral lemma , it can be shown that complex differentiability alone - i.e. without the additional assumption of continuity of the derivatives! - Cauchy's integral theorem and then also the existence of all higher derivatives result. This approach to Cauchy's integral theorem bypasses Stokes' theorem and is preferable from a didactic point of view.

literature

• Kurt Endl, Wolfgang Luh : Analysis. Volume 3: Function Theory, Differential Equations. 6th revised edition. Aula-Verlag, Wiesbaden 1987, ISBN 3-89104-456-9 , p. 143, sentence 4.7.3
• Wolfgang Fischer, Ingo Lieb : Function theory. 7th improved edition. Vieweg, Braunschweig a. a. 1994, ISBN 3-528-67247-1 , p. 57, chapter 3, sentence 1.4 ( Vieweg study. Advanced course in mathematics 47).
• Günter Bärwolff : Higher Mathematics for Natural Scientists and Engineers. 2nd edition, 1st corrected reprint. Spectrum Academic Publishing House, Munich a. a. 2009, ISBN 978-3-8274-1688-9 .
• Klaus Jänich : Introduction to Function Theory . 2nd Edition. Springer-Verlag, Berlin (inter alia) 1980, ISBN 3-540-10032-6 .

Individual evidence

1. ^ Klaus Jänich : Introduction to Function Theory . 2nd Edition. Springer-Verlag, Berlin (inter alia) 1980, ISBN 3-540-10032-6 , pp. 19-20 .
2. ^ Klaus Jänich : Introduction to Function Theory . 2nd Edition. Springer-Verlag, Berlin (inter alia) 1980, ISBN 3-540-10032-6 , pp. 15, 20 .
3. ^ Klaus Jänich : Introduction to Function Theory . 2nd Edition. Springer-Verlag, Berlin (inter alia) 1980, ISBN 3-540-10032-6 , pp. 16, 20 .