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## C. Results from Analytic Function Theory

### C.7 Integrals Revisited

Theorem C.7 (Cauchy Integral Theorem)   If is analytic in some simply connected domain , then is independent of path in and

 (C.7.1)

where is a simple closed path in .

##### Proof

This follows from the Cauchy-Riemann conditions together with Theorem C.2.

We are also interested in the value of integrals in various limiting situations. The following examples cover relevant cases.

We note that if is the length of a simple curve , then

 (C.7.2)

Example C.5  Assume that is a semicircle centered at the origin and having radius . The path length is then . Hence,

• if varies as , then on must vary as - hence, the integral on vanishes for .

• if varies as , then on must vary as - then, the integral on becomes a constant as .

Example C.6   Consider the function and an arc of a circle, , described by for . Then

 (C.7.3)

This is proven as follows. On , we have that . Then

 (C.7.4)

We then use the fact that , and the result follows.

Example C.7  Consider the function

 (C.7.5)

and a semicircle, , defined by for . Then, if is followed clockwise,

 (C.7.6)

This is proven as follows.

On , we have that ; then

 (C.7.7)

We also know that

 (C.7.8)

Then

 (C.7.9)

From this, by evaluation for and for , the result follows.

Example C.8   Consider the function

 (C.7.10)

and a semicircle, , defined by for . Then, for clockwise ,

 (C.7.11)

This is proven as follows.

On , we have that ; then

 (C.7.12)

We recall that, if is a positive real number and , then

 (C.7.13)

Moreover, for very large , we have that

 (C.7.14)

Thus, in the limit, this quantity goes to zero for all positive . The result then follows.

Example C.9  Consider the function

 (C.7.15)

and a semicircle, , defined by for . Then, for clockwise ,

 (C.7.16)

This result is obtained by noting that

 (C.7.17)

and then applying the result in Example C.7.

Example C.10  Consider a function of the form

 (C.7.18)

and , an arc of circle for . Thus, , and

 (C.7.19)

Thus, as , we have that

 (C.7.20)

Example C.11  Consider, now, . If the path is a full circle, centered at the origin and of radius , then

 (C.7.21) (C.7.22)

We can now develop Cauchy's Integral Formula.

Say that can be expanded as

 (C.7.23)

the is called the residue of at .

Consider the path shown in Figure C.3. Because is analytic in a region containing , we have that the integral around the complete path shown in Figure C.3 is zero. The integrals along and cancel. The anticlockwise circular integral around can be computed by following Example C.11 to yield . Hence, the integral around the outer curve is minus the integral around the circle of radius . Thus,

 (C.7.24)

This leads to the following result.

Theorem C.8 (Cauchy's Integral Formula)  Let be analytic in a region. Let be a point inside the region. Then has residue at , and the integral around any closed contour enclosing in a clockwise direction is given by

 (C.7.25)

We note that the residue of at an interior point, , of a region can be obtained by integrating on the boundary of . Hence, we can determine the value of an analytic function inside a region by its behaviour on the boundary.

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