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 .
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
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 :
![D \ subseteq {\ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/819134317effd77a876457d774c8a9ba75d58ea1)
![f \ colon D \ to {\ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa987fddd180b13b032cb4d2d90ef288eb20697)
![a \ in D](https://wikimedia.org/api/rest_v1/media/math/render/svg/227a100dcc33dc54f1033c9e41f1b1a09dc31260)
![D.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6)
![U: = U_ {r} (a) \ subset D](https://wikimedia.org/api/rest_v1/media/math/render/svg/f44d5aeb0d7d3ad2328b2a7b3df3e7b7bd6756cc)
![D.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6)
![z \ in U_ {r} (a)](https://wikimedia.org/api/rest_v1/media/math/render/svg/21e2d609bad423dc6b6a8b02e94c76a7158eeb8e)
![z](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98)
![| za | <r](https://wikimedia.org/api/rest_v1/media/math/render/svg/b27d65e73520e4f32cd74407c865fdc83f3deb2e)
![f (z) = {\ frac {1} {2 \ pi {\ mathrm {i}}}} \ oint _ {{\ partial U}} {\ frac {f (\ zeta)} {\ zeta -z} } {\ mathrm {d}} \ zeta](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cfe2112027384830f4158693e3b2853d5d78fa3)
It is the positively oriented curve for over the top of .
![\ partial U](https://wikimedia.org/api/rest_v1/media/math/render/svg/1360e3416408f1420e852179eb2552c020599426)
![t \ mapsto a + re ^ {{{\ mathrm {i}} t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7a36ab9b855656840e28a96f80822f9f2a9ed75)
![t \ in [0.2 \ pi]](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dbc9ed8510c75442ce1d2e73f021258fc7e04c6)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
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
![z \ in U](https://wikimedia.org/api/rest_v1/media/math/render/svg/60ddfa0dda66703a57db65136203350c952e1feb)
![g \ colon U \ to {\ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/118443ad5bc46c050abfad4bc003793486dd6b75)
![w \ mapsto {\ tfrac {f (w) -f (z)} {wz}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/046b9ef3a24b6220b70bab89a71b942f34c1f0f8)
![w \ neq z](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a005d71923c25d7bf47a969a5dac505868cc93a)
![w \ mapsto f '(z)](https://wikimedia.org/api/rest_v1/media/math/render/svg/27cd987ab26373c9319d867ffc70d77c7ca65c87)
![w = z](https://wikimedia.org/api/rest_v1/media/math/render/svg/60cd1ba897d1f49c5a6e8c0a911cf2ea1519ea96)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
![U \ setminus \ {z \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/782e115782cc0fd3c08b3a04bd698601b801efd1)
-
.
The function , is holomorphic with the derivative , which vanishes because the integrand has an antiderivative (namely ). So is constant, and because of is .
![h \ colon U \ to {\ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e7d98439dc3a7bc45567f1dc103b80461cf795b)
![\ textstyle w \ mapsto \ oint _ {{\ partial U}} {\ tfrac {{\ mathrm {d}} \ zeta} {\ zeta -w}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bdb763b0f307e9d22a008e718eed392c02fd728)
![\ textstyle h '(w) = \ oint _ {{\ partial U}} {\ frac {{\ mathrm {d}} \ zeta} {\ left (\ zeta -w \ right) ^ {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80b4c59a9e70e2ba9a09036467b369324cb73de0)
![\ zeta \ mapsto - {\ tfrac {1} {\ zeta -w}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04b51bd9572ba80cbc8f42dd65918fc056233188)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a)
![h (a) = 2 \ pi {\ mathrm {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce17f42742c2fd0d276b0cc3f99a3560e3aa637)
![h (z) = 2 \ pi {\ mathrm {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bc3410916155feca47836332d626595852d64de)
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 .
![\ zeta (t) = a + re ^ {{{\ mathrm {i}} t}} \ ,, \ {\ mathrm {d}} \ zeta = {\ mathrm {i}} re ^ {{{\ mathrm {i}} t}} {\ mathrm {d}} t](https://wikimedia.org/api/rest_v1/media/math/render/svg/c712f7ba932f60e95a67ae04fdd4aafe4a012db1)
![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](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c12fb2ba8370f5a675e976b3f93384ace6058df)
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 :
![| za | <r](https://wikimedia.org/api/rest_v1/media/math/render/svg/b27d65e73520e4f32cd74407c865fdc83f3deb2e)
![n \ in \ mathbb {N} _ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf5c8ad993619a325ca57a25c22cdc75a460f88)
![f ^ {{(n)}} (z) = {\ frac {n!} {2 \ pi {\ mathrm {i}}}} \ oint _ {{\ partial U}} {\ frac {f (\ zeta)} {\ left (\ zeta -z \ right) ^ {{n + 1}}}} {\ mathrm {d}} \ zeta.](https://wikimedia.org/api/rest_v1/media/math/render/svg/5945186a1d4cba29cf40c860823d0781593ffb50)
Every holomorphic function is locally expandable into a power series for .
![| za | <r](https://wikimedia.org/api/rest_v1/media/math/render/svg/b27d65e73520e4f32cd74407c865fdc83f3deb2e)
![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}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/50735c124141fb38aec0b82f1d00c1629e902ae0)
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:
![f ^ {(n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9dfb1963ccde0e87eb3838f51dc19041e2ff3816)
![on}](https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31)
![| f (z) | \ leq M](https://wikimedia.org/api/rest_v1/media/math/render/svg/1344af5ae232eb4e76f64c8e19b2d01f80727c5d)
![| za | <r \ \ Leftrightarrow z \ in U _ {{r}} (a)](https://wikimedia.org/api/rest_v1/media/math/render/svg/f71a6a54a7e2502f986ea00aed1ef10b241c8835)
![| a _ {{n}} | \ leq {\ frac {M} {r ^ {{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52a831dd9261990f8be452e4b48712e0687d3dc2)
The set of Liouville (each in a very holomorphic
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).
![\ mathbb {C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
proofs
Cauchy's integral formula is partially differentiated, whereby differentiation and integration can be exchanged:
![{\ 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f500daa0491ef141a9234734b07a200380b12bab)
Development of in Cauchy's integral formula with the help of the geometric series results
![{\ frac {1} {\ zeta -z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4df4bb4b477eb871242b9eb054ef54127180223b)
![{\ 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c03d41ca97b208ea55891927e42d2fd14a5baf17)
Since converges uniformly for the geometric series, one can integrate term by term, i.e. H. Swap sum and integral. The expansion coefficients are:
![| za | <| \ zeta -a | = r](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f0f70ead0052dbb1ccd2ccd7d855660b7e7b23c)
![{\ 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29b4179b16065969a27bf4e8d01869812180d83c)
The following estimate applies to the coefficients. There is a with for ; then applies to :
![M> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7f423ab77b3411ec2803520a07c0dfae6ceb826)
![| f (z) | \ leq M](https://wikimedia.org/api/rest_v1/media/math/render/svg/1344af5ae232eb4e76f64c8e19b2d01f80727c5d)
![| za | = r](https://wikimedia.org/api/rest_v1/media/math/render/svg/8aed0fb8cd0cdad0102aa72d271bb5d5cde1117a)
![n \ in \ mathbb {N} _ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf5c8ad993619a325ca57a25c22cdc75a460f88)
![| 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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7aa6fcca8f9ebd6a5a5a64251f11b70523c8374)
Is completely holomorphic and restricted, i.e. for all , then as before for all :
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![\ mathbb {C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
![| f (z) | = | \ sum _ {{n = 0}} ^ {{\ infty}} a _ {{n}} z ^ {{n}} | \ leq M](https://wikimedia.org/api/rest_v1/media/math/render/svg/518d937bbac436dc535f3e3b1c8427979359afe1)
![z \ in {\ mathbb C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/169fae60c23a2027ece2aa7fd4b5047492887e91)
![r> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/23cbbcd53bd13620bc53490e3eec42790850b452)
![| a _ {{n}} | \ leq {\ frac {M} {r ^ {{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52a831dd9261990f8be452e4b48712e0687d3dc2)
Since was arbitrary, then applies to all . Thus it follows from the limitedness of :
![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
![a_ {n} = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a6dcbd7dfa3904fbbfef7745ab8b19904ccf009)
![n \ in \ mathbb {N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![f (z) = a_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88f00507a4770d72de4ec398da8f728b74a21f8c)
That is, every function that is bounded to be entirely holomorphic is constant ( Liouville's theorem ).
![\ mathbb {C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
example
Integrals can also be calculated using the integral formula:
![\ 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7569bccc54acdfabf019bd4f65c6d283000bd04)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258)
![U_ {1}, \ ldots, U_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdefcb9e1f2a3b58c4bb3925fa1d3ed76215b3e7)
![{\ displaystyle \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
![\ textstyle U: = \ prod _ {{i = 1}} ^ {n} U_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1eccc36b9ac88281cca7ec82f365c1e636975f5c)
![{\ displaystyle \ mathbb {C} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258)
![{\ displaystyle f \ colon U \ to \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97128887c8e8a0fac3a7a8a8775682f0680fcfb1)
![\ xi \ in U.](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d143d195651a7007c564bbe8bb2e70ea38d0325)
![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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fae481658aea05f76eedf47f7ee491d8f183dfa7)
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
-
,
with shortened. The formula also applies to multidimensional
![\ partial U = \ partial U_ {1} \ times \ cdots \ times \ partial U_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03a87cc774f2c29fbf6757d7c8fcdbd31eb7d565)
![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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d44b7ec68952e025355a6480588980d840831f08)
for the derivatives of the holomorphic function as well as the Cauchy's inequality
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![\ left | D ^ {k} f (z) \ right | \ leq {\ frac {M \ cdot k!} {r ^ {k}}},](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dc0d069d20cfd9d24205b0896bdc1d398df5b8d)
where and is the radius of the poly cylinder . Another generalization of this integral formula is the Bochner-Martinelli formula .
![\ textstyle M: = \ max _ {{\ xi \ in U}} | f (\ xi) |](https://wikimedia.org/api/rest_v1/media/math/render/svg/413e080343b5dde5e58e12966f2e5597c85e90db)
![r = (r_ {1}, \ ldots, r_ {n})](https://wikimedia.org/api/rest_v1/media/math/render/svg/816b77917dfab5c208200350a32819aaee25590d)
![\ textstyle U: = \ prod _ {{i = 1}} ^ {n} U_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1eccc36b9ac88281cca7ec82f365c1e636975f5c)
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 :
![D \ subseteq {\ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/819134317effd77a876457d774c8a9ba75d58ea1)
![f \ colon D \ to {\ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa987fddd180b13b032cb4d2d90ef288eb20697)
![\Gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19)
![D.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6)
![z \ in D](https://wikimedia.org/api/rest_v1/media/math/render/svg/66c9016b90a20d376dbafef2db16f8dfc177a489)
![\Gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19)
![\ operatorname {ind} _ {{\ Gamma}} (z) f (z) = {\ frac {1} {2 \ pi {\ mathrm {i}}}} \ int _ {\ Gamma} {\ frac { f (\ zeta)} {\ zeta -z}} {\ mathrm {d}} \ zeta](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e9b4486f833c3fb338fd23739d31ae186bd4f7)
Here referred to the winding number of order .
![\ operatorname {ind} _ {{\ Gamma}} (z)](https://wikimedia.org/api/rest_v1/media/math/render/svg/2658a739f2efe602acc8df6ff4c62d96836cf616)
![\Gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19)
![z](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98)
Individual evidence
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^ 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).