2.2 Integration along a contour

Consider the path normal cap gamma from 0 to i in Figure 17, with parametrisation gamma colon left square bracket zero comma three right square bracket long right arrow double-struck cap c given by

gamma of t equals case statement case 1column 1 comma times times two t comma comma less than or equals less than or equals less than or equals zero t one comma case 2column 1 comma plus plus two times times i left parenthesis right parenthesis minus minus t one comma comma less than or equals less than or equals less than or equals one t two comma case 3column 1 comma minus minus plus plus two i times times two left parenthesis right parenthesis minus minus t two comma less than or equals less than or equals less than or equals two t three full stop

This path is not smooth, because gamma is not differentiable at t equals one or t equals two. However, normal cap gamma can be split into three smooth straight-line paths, joined end to end. This leads to the idea of a contour: it is simply what we get when we place a finite number of smooth paths end to end.

Figure 17 A path normal cap gamma from zero to i

Definitions

A contour normal cap gamma is a path that can be subdivided into a finite number of smooth paths normal cap gamma sub one comma normal cap gamma sub two comma ellipsis comma normal cap gamma sub n joined end to end. The order of these constituent smooth paths is indicated by writing

normal cap gamma equals sum with variable number of summands normal cap gamma sub one plus normal cap gamma sub two plus ellipsis plus normal cap gamma sub n full stop

The initial point of normal cap gamma is the initial point of normal cap gamma sub one, and the final point of normal cap gamma is the final point of normal cap gamma sub n.

The definition of a contour is illustrated in Figure 18.

Figure 18 The contour normal cap gamma equals sum with 4 summands normal cap gamma sub one plus normal cap gamma sub two plus normal cap gamma sub three plus normal cap gamma sub four

As an example, the contour normal cap gamma in Figure 17 can be written as sum with 3 summands normal cap gamma sub one plus normal cap gamma sub two plus normal cap gamma sub three, where normal cap gamma sub one, normal cap gamma sub two and normal cap gamma sub three are smooth paths with smooth parametrisations

multiline equation row 1 Blank gamma sub one of t equals two times t left parenthesis t element of left square bracket zero comma one right square bracket right parenthesis comma row 2 Blank gamma sub two of t equals two plus i times left parenthesis t minus one right parenthesis left parenthesis t element of left square bracket one comma two right square bracket right parenthesis comma row 3 Blank gamma sub three of t equals two plus i minus two times left parenthesis t minus two right parenthesis left parenthesis t element of left square bracket two comma three right square bracket right parenthesis full stop
Equation label:(equation 3)

Now, we have seen how to integrate a continuous function along a smooth path. It is natural to extend this definition to contours, by splitting the contour into smooth paths and integrating along each in turn. We formalise this idea in the following definition.

Definition

Let normal cap gamma equals sum with variable number of summands normal cap gamma sub one plus normal cap gamma sub two plus ellipsis plus normal cap gamma sub n be a contour, and let f be a function that is continuous on normal cap gamma. Then the (contour) integral of bold-italic f along bold cap gamma, denoted by integral over normal cap gamma f of z d z, is

integral over normal cap gamma f of z d z equals sum with variable number of summands integral over normal cap gamma sub one f of z d z plus integral over normal cap gamma sub two f of z d z plus ellipsis plus integral over normal cap gamma sub n f of z d z full stop

Remarks

  1. It is clear that a contour can be split into smooth paths in many different ways. Fortunately, all such splittings lead to the same value for the contour integral. We omit the proof of this result, as it is straightforward but tedious.

  2. When evaluating an integral along a contour normal cap gamma equals sum with variable number of summands normal cap gamma sub one plus normal cap gamma sub two plus ellipsis plus normal cap gamma sub n, we often consider each smooth path normal cap gamma sub one comma normal cap gamma sub two comma ellipsis comma normal cap gamma sub n separately, using a convenient parametrisation in each case. For example, consider the contour normal cap gamma equals sum with 3 summands normal cap gamma sub one plus normal cap gamma sub two plus normal cap gamma sub three of Figure 17. To evaluate a contour integral of the form

    integral over normal cap gamma f of z d z equals sum with 3 summands integral over normal cap gamma sub one f of z d z plus integral over normal cap gamma sub two f of z d z plus integral over normal cap gamma sub three f of z d z comma

    we can use the smooth parametrisations (above) of normal cap gamma sub one, normal cap gamma sub two and normal cap gamma sub three, or we could use another convenient choice of parametrisations, such as

    multiline equation row 1 gamma sub one of t equals t left parenthesis t element of left square bracket zero comma two right square bracket right parenthesis comma row 2 gamma sub two of t equals two plus i times t left parenthesis t element of left square bracket zero comma one right square bracket right parenthesis comma row 3 gamma sub three of t equals two plus i minus t left parenthesis t element of left square bracket zero comma two right square bracket right parenthesis full stop

  3. The alternative notation integral over normal cap gamma f is sometimes used for contour integrals when the omission of the integration variable z will cause no confusion.

Example 5

Evaluate

integral over normal cap gamma z squared d z comma

where normal cap gamma is the contour shown in Figure 19.

Figure 19 A contour normal cap gamma from zero to one plus i

Solution

We split normal cap gamma into two smooth paths normal cap gamma equals normal cap gamma sub one plus normal cap gamma sub two, where normal cap gamma sub one is the line segment from 0 to 1 with parametrisation gamma sub one of t equals t left parenthesis t element of left square bracket zero comma one right square bracket right parenthesis, and normal cap gamma sub two is the line segment from 1 to one plus i, with parametrisation gamma sub two of t equals one plus i times t left parenthesis t element of left square bracket zero comma one right square bracket right parenthesis. Then

multiline equation row 1 integral over normal cap gamma z squared d z equals integral over normal cap gamma sub one z squared d z plus integral over normal cap gamma sub two z squared d z row 2 Blank equals integral over zero under one t squared d t plus integral over zero under one left parenthesis one plus i times t right parenthesis squared times i d t row 3 Blank equals integral over zero under one t squared d t plus integral over zero under one left parenthesis negative two times t plus i minus i times t squared right parenthesis d t row 4 Blank equals integral over zero under one left parenthesis t squared minus two times t right parenthesis d t plus i times integral over zero under one left parenthesis one minus t squared right parenthesis d t row 5 Blank equals left square bracket one divided by three times t cubed minus t squared right square bracket sub zero super one plus i times left square bracket t minus one divided by three times t cubed right square bracket sub zero super one row 6 Blank equals left parenthesis one divided by three minus one right parenthesis plus i times left parenthesis one minus one divided by three right parenthesis row 7 Blank equals negative two divided by three plus two divided by three times i full stop

Notice that this answer is the same as that obtained in Example 2 for

integral over normal cap gamma z squared d z comma

where normal cap gamma is the line segment from zero to one plus i. The reason for this will become clear when we get to Theorem 8, the Contour Independence Theorem.

Exercise 5

Evaluate

integral over normal cap gamma z macron d z

for each of the following contours normal cap gamma.

In part (b) the contour consists of a line segment and a semicircle, traversed once anticlockwise. Take negative one to be the initial (and final) point of this contour.

Answer

  1. normal cap gamma equals sum with 3 summands normal cap gamma sub one plus normal cap gamma sub two plus normal cap gamma sub three, where normal cap gamma sub one is the line segment from 0 to 1, normal cap gamma sub two is the line segment from 1 to one plus i, and normal cap gamma sub three is the line segment from one plus i to i. We choose to use the associated standard parametrisations

    multiline equation row 1 Blank gamma sub one of t equals t times left parenthesis t element of left square bracket zero comma one right square bracket right parenthesis comma row 2 Blank gamma sub two of t equals one plus i times t times left parenthesis t element of left square bracket zero comma one right square bracket right parenthesis comma row 3 Blank gamma sub three of t equals one minus t plus i times left parenthesis t element of left square bracket zero comma one right square bracket right parenthesis full stop

    Then gamma times sub one super prime times left parenthesis t right parenthesis equals one, gamma times sub two super prime times left parenthesis t right parenthesis equals i, gamma times sub three super prime times left parenthesis t right parenthesis equals negative one. Hence

    multiline equation row 1 integral over normal cap gamma z macron d z equals sum with 3 summands integral over normal cap gamma sub one z macron d z plus integral over normal cap gamma sub two z macron d z plus integral over normal cap gamma sub three z macron d z row 2 Blank equals integral over zero under one t multiplication one d t plus integral over zero under one left parenthesis one minus i times t right parenthesis multiplication i d t row 3 Blank prefix plus of integral over zero under one left parenthesis one minus t minus i right parenthesis multiplication left parenthesis negative one right parenthesis d t row 4 Blank equals integral over zero under one left parenthesis three times t plus two times i minus one right parenthesis d t row 5 Blank equals left square bracket three divided by two times t squared plus left parenthesis two times i minus one right parenthesis times t right square bracket sub zero super one row 6 Blank equation sequence part 1 equals part 2 three divided by two plus two times i minus one equals part 3 one divided by two plus two times i full stop

  2. normal cap gamma equals normal cap gamma sub one plus normal cap gamma sub two, where normal cap gamma sub one is the line segment from negative one to 1, and normal cap gamma sub two is the upper half of the circle with centre 0 from 1 to negative one. We choose to use the parametrisations

    multiline equation row 1 Blank gamma sub one of t equals t times left parenthesis t element of left square bracket negative one comma one right square bracket right parenthesis comma row 2 Blank gamma sub two of t equals e super i times t times left parenthesis t element of left square bracket zero comma pi right square bracket right parenthesis full stop

    Then gamma times sub one super prime times left parenthesis t right parenthesis equals one, gamma times sub two super prime times left parenthesis t right parenthesis equals i times e super i times t. Hence

    multiline equation row 1 integral over normal cap gamma z macron d z equals integral over normal cap gamma sub one z macron d z plus integral over normal cap gamma sub two z macron d z row 2 Blank equals integral over negative one under one t multiplication one d t plus integral over zero under pi e super negative i times t multiplication i times e super i times t d t row 3 Blank equals integral over negative one under one t d t plus i times integral over zero under pi one d t row 4 Blank equals left square bracket one divided by two times t squared right square bracket sub negative one super one plus i times left square bracket t right square bracket sub zero super pi row 5 Blank equation sequence part 1 equals part 2 zero plus i times pi equals part 3 pi times i full stop

This section will conclude by stating some rules for combining contour integrals. To prove them, we split the contour normal cap gamma into constituent smooth paths, and use the Sum Rule and Multiple Rule for real integration given in Theorem 3 to prove the results for each path. We omit the details.

Theorem 5 Combination Rules for Contour Integrals

Let normal cap gamma be a contour, and let f and g be functions that are continuous on normal cap gamma.

  1. Sum Rule integral over normal cap gamma left parenthesis f of z plus g of z right parenthesis d z equals integral over normal cap gamma f of z d z plus integral over normal cap gamma g of z d z full stop

  2. Multiple Rule integral over normal cap gamma lamda times f of z d z equals lamda times integral over normal cap gamma f of z d z comma where lamda element of double-struck cap c full stop

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