Cauchy's integral formula

The Cauchy integral formula (after Augustin Louis Cauchy ) is one of the fundamental statements of function theory , a branch of mathematics . In its weakest form, it says that the values ​​of a holomorphic function inside a circular disk are already determined by their values ​​on the edge of this circular disk. A strong generalization of this is the residual theorem . ${\displaystyle f}$

Cauchy's integral formula for circular disks

statement

Is open, holomorphic, a point in and a relatively compact circular disk in , then applies to all , i.e. to all with : ${\displaystyle D\subseteq \mathbb {C} }$ ${\displaystyle f\colon D\to \mathbb {C} }$ ${\displaystyle a\in D}$ ${\displaystyle D}$ ${\displaystyle U:=U_{r}(a)\subset D}$ ${\displaystyle D}$ ${\displaystyle z\in U_{r}(a)}$ ${\displaystyle z}$ ${\displaystyle |z-a|<r}$

${\displaystyle f(z)={\frac {1}{2\pi \mathrm {i} }}\oint _{\partial U}{\frac {f(\zeta )}{\zeta -z}}\mathrm {d} \zeta }$

It is the positively oriented curve for over the top of . ${\displaystyle \partial U}$ ${\displaystyle t\mapsto a+re^{\mathrm {i} t}}$ ${\displaystyle t\in [0,2\pi ]}$ ${\displaystyle U}$

proof

For solid the function is defined by for and for . is steady on and holomorphic on . With Cauchy's integral theorem, we now have ${\displaystyle z\in U}$ ${\displaystyle g\colon U\to \mathbb {C} }$ ${\displaystyle w\mapsto {\tfrac {f(w)-f(z)}{w-z}}}$ ${\displaystyle w\neq z}$ ${\displaystyle w\mapsto f'(z)}$ ${\displaystyle w=z}$ ${\displaystyle g}$ ${\displaystyle U}$ ${\displaystyle U\setminus \{z\}}$

${\displaystyle 0=\oint _{\partial U}g=\oint _{\partial U}{\frac {f(\zeta )}{\zeta -z}}\mathrm {d} \zeta -f(z)\oint _{\partial U}{\frac {\mathrm {d} \zeta }{\zeta -z}}}$.

The function , is holomorphic with the derivative , which vanishes because the integrand has an antiderivative (namely ). So is constant, and because of is . ${\displaystyle h\colon U\to \mathbb {C} }$ ${\displaystyle \textstyle w\mapsto \oint _{\partial U}{\tfrac {\mathrm {d} \zeta }{\zeta -w}}}$ ${\displaystyle \textstyle h'(w)=\oint _{\partial U}{\frac {\mathrm {d} \zeta }{\left(\zeta -w\right)^{2}}}}$ ${\displaystyle \zeta \mapsto -{\tfrac {1}{\zeta -w}}}$ ${\displaystyle h}$ ${\displaystyle h(a)=2\pi \mathrm {i} }$ ${\displaystyle h(z)=2\pi \mathrm {i} }$

Inferences

The following applies to every holomorphic function: The function value in the center of a circle is the mean value of the function values ​​on the edge of the circle. Use it . ${\displaystyle \zeta (t)=a+re^{\mathrm {i} t}\,,\ \mathrm {d} \zeta =\mathrm {i} re^{\mathrm {i} t}\mathrm {d} t}$

${\displaystyle f|_{U}(a)={\frac {1}{2\pi \mathrm {i} }}\oint _{\partial U}{\frac {f(\zeta )}{\zeta -a}}\mathrm {d} \zeta ={\frac {1}{2\pi \mathrm {i} }}\int _{0}^{2\pi }{\frac {f(a+re^{\mathrm {i} t})}{re^{\mathrm {i} t}}}\mathrm {i} re^{\mathrm {i} t}\,\mathrm {d} t={\frac {1}{2\pi }}\int _{0}^{2\pi }f(a+re^{\mathrm {i} t})\,\mathrm {d} t}$

Every holomorphic function can be complexly differentiated as often as desired and each of these derivatives is holomorphic. Expressed with the integral formula this means for and : ${\displaystyle |z-a|<r}$ ${\displaystyle n\in \mathbb {N} _{0}}$

${\displaystyle f^{(n)}(z)={\frac {n!}{2\pi \mathrm {i} }}\oint _{\partial U}{\frac {f(\zeta )}{\left(\zeta -z\right)^{n+1}}}\mathrm {d} \zeta .}$

Every holomorphic function is locally expandable into a power series for . ${\displaystyle |z-a|<r}$

${\displaystyle f(z)=\sum \limits _{n=0}^{\infty }\left({\frac {1}{2\pi \mathrm {i} }}\oint _{\partial U}{\frac {f(\zeta )}{\left(\zeta -a\right)^{n+1}}}\mathrm {d} \zeta \right)(z-a)^{n}=\sum \limits _{n=0}^{\infty }a_{n}(z-a)^{n}.}$

With the integral formula for it immediately follows that the coefficients are exactly the Taylor coefficients . For the coefficients following estimate applies if, for the following applies: ${\displaystyle f^{(n)}}$ ${\displaystyle a_{n}}$ ${\displaystyle |f(z)|\leq M}$ ${\displaystyle |z-a|<r\ \Leftrightarrow z\in U_{r}(a)}$

${\displaystyle |a_{n}|\leq {\frac {M}{r^{n}}}}$

The set of Liouville (each in a very holomorphic ${\displaystyle \mathbb {C} }$ bounded function is constant) can very quickly show the integral formula. With this one can easily prove the fundamental theorem of algebra (every polynomial splits into linear factors). ${\displaystyle \mathbb {C} }$

proofs

Cauchy's integral formula is partially differentiated, whereby differentiation and integration can be exchanged:

${\displaystyle {\begin{aligned}f^{(n)}|_{U}(z)&={\frac {\partial ^{n}f}{\partial z^{n}}}|_{U}(z)={\frac {1}{2\pi \mathrm {i} }}{\frac {\partial ^{n}}{\partial z^{n}}}\oint _{\partial U}{\frac {f(\zeta )}{\zeta -z}}\mathrm {d} \zeta \\&={\frac {1}{2\pi \mathrm {i} }}\oint _{\partial U}f(\zeta )\underbrace {{\frac {\partial ^{n}}{\partial z^{n}}}{\frac {1}{\zeta -z}}} _{n!/(\zeta -z)^{1+n}}\mathrm {d} \zeta ={\frac {n!}{2\pi \mathrm {i} }}\oint _{\partial U}{\frac {f(\zeta )}{(\zeta -z)^{1+n}}}\mathrm {d} \zeta \end{aligned}}}$

Development of in Cauchy's integral formula with the help of the geometric series results ${\displaystyle {\frac {1}{\zeta -z}}}$

${\displaystyle {\begin{aligned}f|_{U}(z)&={\frac {1}{2\pi \mathrm {i} }}\oint _{\partial U_{r}(a)}{\frac {f(\zeta )}{\zeta -z}}\mathrm {d} \zeta ={\frac {1}{2\pi \mathrm {i} }}\oint _{\partial U_{r}(a)}{\frac {f(\zeta )}{\zeta -a-(z-a)}}\mathrm {d} \zeta \\&={\frac {1}{2\pi \mathrm {i} }}\oint _{\partial U_{r}(a)}{\frac {f(\zeta )}{\zeta -a}}{\frac {1}{1-{\frac {z-a}{\zeta -a}}}}\mathrm {d} \zeta \,{\overset {|{\frac {z-a}{\zeta -a}}|<1}{=}}\,{\frac {1}{2\pi \mathrm {i} }}\oint _{\partial U_{r}(a)}{\frac {f(\zeta )}{\zeta -a}}\sum _{n=0}^{\infty }\left({\frac {z-a}{\zeta -a}}\right)^{n}\mathrm {d} \zeta \\&=\sum _{n=0}^{\infty }\underbrace {\left({\frac {1}{2\pi \mathrm {i} }}\oint _{\partial U_{r}(a)}{\frac {f(\zeta )}{(\zeta -a)^{n+1}}}\mathrm {d} \zeta \right)} _{a_{n}}(z-a)^{n}\end{aligned}}}$

Since converges uniformly for the geometric series, one can integrate term by term, i.e. H. Swap sum and integral. The expansion coefficients are: ${\displaystyle |z-a|<|\zeta -a|=r}$

${\displaystyle {\begin{aligned}a_{n}&={\frac {1}{n!}}f^{(n)}|_{U}(a)={\frac {1}{2\pi \mathrm {i} }}\oint _{\partial U_{r}(a)}{\frac {f(\zeta )}{(\zeta -a)^{n+1}}}\mathrm {d} \zeta \\&={\frac {1}{2\pi \mathrm {i} }}\int _{0}^{2\pi }{\frac {f(a+re^{\mathrm {i} t})}{(re^{\mathrm {i} t})^{n+1}}}\mathrm {i} re^{\mathrm {i} t}\,\mathrm {d} t={\frac {1}{2\pi r^{n}}}\int _{0}^{2\pi }f(a+re^{\mathrm {i} t})e^{-\mathrm {i} nt}\,\mathrm {d} t\end{aligned}}}$

The following estimate applies to the coefficients. There is a with for ; then applies to : ${\displaystyle M>0}$ ${\displaystyle |f(z)|\leq M}$ ${\displaystyle |z-a|=r}$ ${\displaystyle n\in \mathbb {N} _{0}}$

${\displaystyle |a_{n}|=\left|{\frac {1}{2\pi r^{n}}}\int _{0}^{2\pi }f(a+re^{\mathrm {i} t})e^{-\mathrm {i} nt}\,\mathrm {d} t\right|\leq {\frac {1}{2\pi r^{n}}}\int _{0}^{2\pi }\underbrace {|f(a+re^{\mathrm {i} t})|} _{\leq M}\,\mathrm {d} t\leq {\frac {M}{r^{n}}}}$

Is completely holomorphic and restricted, i.e. for all , then as before for all : ${\displaystyle f}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle |f(z)|=|\sum _{n=0}^{\infty }a_{n}z^{n}|\leq M}$ ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle r>0}$

${\displaystyle |a_{n}|\leq {\frac {M}{r^{n}}}}$

Since was arbitrary, then applies to all . Thus it follows from the limitedness of : ${\displaystyle r}$ ${\displaystyle a_{n}=0}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle f}$

${\displaystyle f(z)=a_{0}}$

That is, every function that is bounded to be entirely holomorphic is constant ( Liouville's theorem ). ${\displaystyle \mathbb {C} }$

example

Integrals can also be calculated using the integral formula:

${\displaystyle \oint _{\partial U_{2}(0)}{\frac {e^{2\zeta }}{\left(\zeta +1\right)^{4}}}\mathrm {d} \zeta ={\frac {2\pi \mathrm {i} }{3!}}{\frac {\mathrm {d} ^{3}}{\mathrm {d} z^{3}}}e^{2z}|_{z=-1}={\frac {8\pi \mathrm {i} }{3e^{2}}}}$

Cauchy's integral formula for poly cylinders

Cauchy's integral formula was also generalized to the multi-dimensional, complex space . If circular disks are in , then a poly cylinder is in . Let be a holomorphic function and then Cauchy's integral formula is through ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle U_{1},\ldots ,U_{n}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \textstyle U:=\prod _{i=1}^{n}U_{i}}$ ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle f\colon U\to \mathbb {C} }$ ${\displaystyle \xi \in U.}$

${\displaystyle f(z_{1},\ldots ,z_{n})={\frac {1}{(2\pi i)^{n}}}\oint _{\partial U_{n}}\cdots \oint _{\partial U_{1}}{\frac {f(\xi _{1},\ldots ,\xi _{n})}{(\xi _{1}-z_{1})\cdots (\xi _{n}-z_{n})}}\mathrm {d} \xi _{1}\cdots \mathrm {d} \xi _{n}}$

explained. Since Cauchy's integral theorem does not hold in multi-dimensional space, this formula cannot be derived from it analogously to the one-dimensional case. This integral formula is therefore derived from Cauchy's integral formula for circular disks with the help of induction . Using the multi-index notation , the formula can be restored to

${\displaystyle f(z)={\frac {1}{(2\pi i)^{n}}}\oint _{\partial U}{\frac {f(\xi )}{(\xi -z)}}\,\mathrm {d} \xi }$,

with shortened. The formula also applies to multidimensional ${\displaystyle \partial U=\partial U_{1}\times \cdots \times \partial U_{n}}$

${\displaystyle D^{k}f(z_{1},\ldots ,z_{n})={\frac {k!}{(2\pi i)^{n}}}\oint _{\partial U_{n}}\cdots \oint _{\partial U_{1}}{\frac {f(\xi _{1},\ldots ,\xi _{n})}{(\xi _{1}-z_{1})^{k_{1}+1}\cdots (\xi _{n}-z_{n})^{k_{n}+1}}}d\xi _{1}\cdots d\xi _{n}}$

for the derivatives of the holomorphic function as well as the Cauchy's inequality ${\displaystyle f}$

${\displaystyle \left|D^{k}f(z)\right|\leq {\frac {M\cdot k!}{r^{k}}},}$

where and is the radius of the poly cylinder . Another generalization of this integral formula is the Bochner-Martinelli formula . ${\displaystyle \textstyle M:=\max _{\xi \in U}|f(\xi )|}$ ${\displaystyle r=(r_{1},\ldots ,r_{n})}$ ${\displaystyle \textstyle U:=\prod _{i=1}^{n}U_{i}}$

Cauchy's integral formula for cycles

The version for cycles represents a generalization of the integral formula for circular curves:

If a region is holomorphic and a zero-homologous cycle in , then the following integral formula applies to all that are not on : ${\displaystyle D\subseteq \mathbb {C} }$ ${\displaystyle f\colon D\to \mathbb {C} }$ ${\displaystyle \Gamma }$ ${\displaystyle D}$ ${\displaystyle z\in D}$ ${\displaystyle \Gamma }$

${\displaystyle \operatorname {ind} _{\Gamma }(z)f(z)={\frac {1}{2\pi \mathrm {i} }}\int _{\Gamma }{\frac {f(\zeta )}{\zeta -z}}\mathrm {d} \zeta }$

Here referred to the winding number of order . ${\displaystyle \operatorname {ind} _{\Gamma }(z)}$ ${\displaystyle \Gamma }$ ${\displaystyle z}$

Individual evidence

1. ^ Lars Hörmander : An Introduction to Complex Analysis in Several Variables. North Holland Pub. Co. et al., Amsterdam et al. 1973, ISBN 0-444-10523-9 , pp. 25-27.

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. 153, sentence 4.9.1.
• Wolfgang Fischer, Ingo Lieb : Function theory. 7th improved edition. Vieweg, Braunschweig et al. 1994, ISBN 3-528-67247-1 , p. 60, chapter 3, sentence 2.2 ( Vieweg study. Advanced course in mathematics 47).