# 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 | za | ${\ displaystyle f (z) = {\ frac {1} {2 \ pi \ mathrm {i}}} \ oint _ {\ partial U} {\ frac {f (\ zeta)} {\ zeta -z}} \ mathrm {d} \ zeta}$ 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 ${\ displaystyle z \ in U}$ ${\ displaystyle g \ colon U \ to \ mathbb {C}}$ ${\ displaystyle w \ mapsto {\ tfrac {f (w) -f (z)} {wz}}}$ ${\ displaystyle w \ neq z}$ ${\ displaystyle w \ mapsto f '(z)}$ ${\ displaystyle w = z}$ ${\ displaystyle g}$ ${\ displaystyle U}$ ${\ displaystyle U \ setminus \ {z \}}$ ${\ displaystyle 0 = \ oint _ {\ partial U} g = \ oint _ {\ partial U} {\ frac {f (\ zeta)} {\ zeta -z}} \ mathrm {d} \ zeta -f ( z) \ oint _ {\ partial U} {\ frac {\ mathrm {d} \ zeta} {\ zeta -z}}}$ .

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 | za | ${\ displaystyle n \ in \ mathbb {N} _ {0}}$ ${\ displaystyle f ^ {(n)} (z) = {\ frac {n!} {2 \ pi \ mathrm {i}}} \ oint _ {\ partial U} {\ frac {f (\ zeta)} {\ left (\ zeta -z \ right) ^ {n + 1}}} \ mathrm {d} \ zeta.}$ Every holomorphic function is locally expandable into a power series for . ${\ displaystyle | za | ${\ displaystyle f (z) = \ sum \ limits _ {n = 0} ^ {\ infty} \ left ({\ frac {1} {2 \ pi \ mathrm {i}}} \ oint _ {\ partial U } {\ frac {f (\ zeta)} {\ left (\ zeta -a \ right) ^ {n + 1}}} \ mathrm {d} \ zeta \ right) (za) ^ {n} = \ sum \ limits _ {n = 0} ^ {\ infty} a_ {n} (za) ^ {n}.}$ 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 | za | ${\ displaystyle | a_ {n} | \ leq {\ frac {M} {r ^ {n}}}}$ 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- (za)}} \ mathrm {d} \ zeta \\ & = {\ frac {1} {2 \ pi \ mathrm {i}}} \ oint _ {\ partial U_ {r} (a)} {\ frac {f (\ zeta)} {\ zeta -a}} {\ frac {1} {1 - {\ frac { za} {\ zeta -a}}}} \ mathrm {d} \ zeta \, {\ overset {| {\ frac {za} {\ 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 {za} {\ 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}} (za) ^ {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: ${\ displaystyle | za | <| \ zeta -a | = r}$ {\ 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 : ${\ displaystyle M> 0}$ ${\ displaystyle | f (z) | \ leq M}$ ${\ displaystyle | za | = r}$ ${\ displaystyle n \ in \ mathbb {N} _ {0}}$ ${\ displaystyle | a_ {n} | = \ left | {\ frac {1} {2 \ pi r ^ {n}}} \ int _ {0} ^ {2 \ pi} f (a + re ^ {\ mathrm {i} t}) e ^ {- \ mathrm {i} nt} \, \ mathrm {d} t \ right | \ leq {\ frac {1} {2 \ pi r ^ {n}}} \ int _ {0} ^ {2 \ pi} \ underbrace {| f (a + re ^ {\ mathrm {i} t}) |} _ {\ leq M} \, \ mathrm {d} t \ leq {\ frac {M} {r ^ {n}}}}$ 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).