corollary of Cauchy integral theorem

Theorem.  Let γ be a closed contour of not intersecting itself and γ1,,γn likewise.  Let all the contours γ1,,γn be outside each other but insinde γ.  If the closed, “holey” domain between γ and the γjs is contained in a domain U where f is holomorphic, then

γf(z)𝑑z=γ1f(z)𝑑z++γnf(z)𝑑z (1)

where all integrals are taken with the same direction of circulation.  Especially, in the case  n=1  one has

γf(z)𝑑z=γ1f(z)𝑑z. (2)

Note 1.  The integrals in (1) and (2) need not necessarily vanish, since inside a γj there may be points not belonging to U.

Note 2.  When  n=0,  i.e. the sum on the right hand side of (1) is empty (, it has the value 0; thus also the Cauchy integral theorem is a special case of (1).

Note 3.  The theorem implies easily the residue theoremMathworldPlanetmath.

Proof.  We prove the theorem only in the case  n=1.  Other cases may be handled analogously.
Draw two auxiliary ways connecting γ and γ1.  The integral of f taken anticlockwise around the route consisting of the upper parts of the curves and the auxiliary ways is, by the fundamental theorem of complex analysis, equal zero.  Similarly the integral of f taken anticlockwise around the route consisting of the lower parts of the curves and the auxiliary ways is equal zero.  Thus also the sum of both equals zero.  But in the sum, the portions taken along the auxiliary ways are run in opposite directions and so they cancel each other.  Therefore in the sum only the portions, which are run along γ and γ1, remain; then

γf(z)𝑑z+γ1f(z)𝑑z= 0,



However, here γ is run anticlockwise and γ1 clockwise.  Reversing the direction in the right hand side of this last equation, one obtains (2). Q.E.D.

Example.  Calculate


where the circle C of complex planeMathworldPlanetmath with centre z0 and radius R is run once anticlockwise.
Since  |z-z0|=R  we can take the parametric presentation


Then  dz=iReiφdφ  and

Cdzz-z0=02πi𝑑φ= 2iπ. (3)

By the equation (2) of the theorem, one can infer that the result (3) is true for any continuousMathworldPlanetmath contour going once around the point z0 anticlockwise (cf. the lemma of entry).

Title corollary of Cauchy integral theorem
Canonical name CorollaryOfCauchyIntegralTheorem
Date of creation 2013-03-22 18:54:12
Last modified on 2013-03-22 18:54:12
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 18
Author pahio (2872)
Entry type Theorem
Classification msc 30E20
Synonym generalisation of Cauchy integral theorem
Related topic VariantOfCauchyIntegralFormula