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Introduction to complex analysis
Introduction to complex analysis

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3.1 The Fundamental Theorem of Calculus

In Example 5 we saw that

integral over normal cap gamma z squared d z equals negative two divided by three plus two divided by three times i comma

where normal cap gamma is the contour shown in Figure 23. Our method was to write down a smooth parametrisation for each of the two line segments, replace z in the integral by these parametrisations, and then integrate.

Described image
Figure 23 A contour normal cap gamma from zero to one plus i

It is, however, tempting to approach this integral as you would a corresponding real integral and write

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

The Fundamental Theorem of Calculus for contour integrals tells us that this method of evaluation is permissible under certain conditions. Before stating it, we need the idea of a primitive of a complex function, which is defined in a similar way to the primitive of a real function (Section 1.3).

Definition

Let f and cap f be functions defined on a region script cap r. Then cap f is a primitive of bold-italic f on bold-script cap r if cap f is analytic on script cap r and

cap f super prime of z equals f of z comma for all z element of script cap r full stop

The function cap f is also called an antiderivative or indefinite integral of f on script cap r.

For example, cap f of z equals one divided by three times z cubed is a primitive of f of z equals z squared on double-struck cap c, since cap f is analytic on double-struck cap c and cap f super prime of z equals z squared, for all z element of double-struck cap c. Another primitive is cap f of z equals one divided by three times z cubed plus two times i; indeed, any function of the form cap f of z equals one divided by three times z cubed plus c, where c element of double-struck cap c, is a primitive of f on double-struck cap c.

Exercise 10

Write down a primitive cap f of each of the following functions f on the given region script cap r.

  • a.f of z equals e super three times i times z comma script cap r equals double-struck cap c

  • b.f of z equals left parenthesis one plus i times z right parenthesis super negative two comma script cap r equals double-struck cap c minus i

  • c.f of z equals z super negative one comma script cap r equals z colon Re of z greater than zero

Answer

  • a.cap f of z equals one divided by three times i times e super three times i times z times left parenthesis z element of double-struck cap c right parenthesis

  • b.equation sequence part 1 cap f of z equals part 2 i times left parenthesis one plus i times z right parenthesis super negative one equals part 3 left parenthesis z minus i right parenthesis super negative one times left parenthesis z element of double-struck cap c minus i right parenthesis

  • c.cap f of z equals Log of z of Re of z greater than zero

We now state the Fundamental Theorem of Calculus for contour integrals, which gives us a quick way of evaluating a contour integral of a function with a primitive that we can determine. The theorem will be proved later in this section.

Theorem 7 Fundamental Theorem of Calculus

Let f be a function that is continuous and has a primitive cap f on a region script cap r, and let normal cap gamma be a contour in script cap r with initial point alpha and final point beta. Then

integral over normal cap gamma f of z d z equals cap f of beta minus cap f of alpha full stop

We often use the notation

left square bracket cap f of z right square bracket sub alpha super beta equals cap f of beta minus cap f of alpha full stop

Some texts write cap f of z vertical line sub alpha super beta instead of left square bracket cap f of z right square bracket sub alpha super beta.

For an example of the use of the Fundamental Theorem of Calculus, observe that if f of z equals z squared, then f is continuous on double-struck cap c and has a primitive cap f of z equals one divided by three times z cubed there. Hence, for the contour normal cap gamma in Figure 23, we can write

equation sequence part 1 integral over normal cap gamma z squared d z equals part 2 left square bracket one divided by three times z cubed right square bracket sub zero super one plus i equals part 3 one divided by three times left parenthesis one plus i right parenthesis cubed minus one divided by three multiplication zero cubed equals part 4 negative two divided by three plus two divided by three times i full stop

Exercise 11

Use the Fundamental Theorem of Calculus to evaluate

integral over normal cap gamma e super three times i times z d z comma

where normal cap gamma is the semicircular path shown in Figure 24.

Described image
Figure 24 A semicircular path normal cap gamma from two to negative two

Answer

Let f of z equals e super three times i times z, cap f of z equals e super three times i times z solidus left parenthesis three times i right parenthesis and script cap r script equals double-struck cap c. Then f is continuous on script cap r, and cap f is a primitive of f on script cap r. Thus, by the Fundamental Theorem of Calculus,

multiline equation row 1 integral over normal cap gamma e super three times i times z d z equals cap f of negative two minus cap f of two row 2 Blank equation sequence part 1 equals part 2 one divided by three times i times left parenthesis e super negative six times i minus e super six times i right parenthesis equals part 3 negative two divided by three times sine of six full stop

The final simplification follows from the formula

sine of z equals one divided by two times i times left parenthesis e super i times z minus e super negative i times z right parenthesis comma

with z equals six.

You have seen that

integral over normal cap gamma z squared d z equals negative two divided by three plus two divided by three times i

both when normal cap gamma is the contour in Figure 23 and also when normal cap gamma is the line segment from zero to one plus i (see Example 2). This is not a coincidence: in fact, it is a particular case of the following important consequence of the Fundamental Theorem of Calculus.

Theorem 8 Contour Independence Theorem

Let f be a function that is continuous and has a primitive cap f on a region script cap r, and let normal cap gamma sub one and normal cap gamma sub two be contours in script cap r with the same initial point alpha and the same final point beta. Then

integral over normal cap gamma sub one f of z d z equals integral over normal cap gamma sub two f of z d z full stop

Proof

By the Fundamental Theorem of Calculus for contour integrals, the value of each of these integrals is cap f of beta minus cap f of alpha.

The idea that a contour integral may, under suitable hypotheses, depend only on the endpoints of the contour (and not on the contour itself) has great significance.

Exercise 12

Use the Fundamental Theorem of Calculus to evaluate the following integrals.

  • a.integral over normal cap gamma e super negative pi times z d z, where normal cap gamma is any contour from negative i to i.

  • b.integral over normal cap gamma left parenthesis three times z minus one right parenthesis squared d z, where normal cap gamma is any contour from 2 to two times i plus one divided by three.

  • c.integral over normal cap gamma hyperbolic sine of z times d times z, where normal cap gamma is any contour from i to 1.

  • d.integral over normal cap gamma e super sine of z times cosine of z times d times z, where normal cap gamma is any contour from 0 to pi solidus two.

  • e.integral over normal cap gamma sine of z divided by cosine squared of z d z, where normal cap gamma is any contour from 0 to pi lying in double-struck cap c minus left parenthesis n plus one divided by two right parenthesis times pi colon n element of double-struck cap z.

Answer

  • a.Let f of z equals e super negative pi times z, cap f of z equals negative e super negative pi times z solidus pi and script cap r equals double-struck cap c. Then f is continuous on script cap r, and cap f is a primitive of f on script cap r. Thus, by the Fundamental Theorem of Calculus,

    multiline equation row 1 integral over normal cap gamma e super negative pi times z d z equals cap f of i minus cap f of negative i row 2 Blank equals left parenthesis negative e super negative pi times i solidus pi right parenthesis minus left parenthesis negative e super pi times i solidus pi right parenthesis row 3 Blank equation sequence part 1 equals part 2 one solidus pi minus one solidus pi equals part 3 zero full stop
  • b.Let f of z equals left parenthesis three times z minus one right parenthesis squared, cap f of z equals one divided by nine times left parenthesis three times z minus one right parenthesis cubed and script cap r equals double-struck cap c. Then f is continuous on script cap r, and cap f is a primitive of f on script cap r. Thus, by the Fundamental Theorem of Calculus,

    multiline equation row 1 integral over normal cap gamma left parenthesis three times z minus one right parenthesis squared d z equals cap f times left parenthesis two times i plus one divided by three right parenthesis minus cap f of two row 2 Blank equals one divided by nine times left parenthesis six times i right parenthesis cubed minus one divided by nine multiplication five cubed row 3 Blank equals negative one divided by nine times left parenthesis 125 plus 216 times i right parenthesis full stop
  • c.Let f of z equals hyperbolic sine of z, cap f of z equals hyperbolic cosine of z and script cap r equals double-struck cap c. Then f is continuous on script cap r, and cap f is a primitive of f on script cap r. Thus, by the Fundamental Theorem of Calculus,

    multiline equation row 1 integral over normal cap gamma hyperbolic sine of z times d times z equals cap f of one minus cap f of i row 2 Blank equals hyperbolic cosine of one minus hyperbolic cosine of i row 3 Blank equals hyperbolic cosine of one minus cosine of one full stop
  • d.The integrand e super sine of z times cosine of z can be written as

    exp of sine of z multiplication sine super prime of z comma

    which equals left parenthesis exp ring operator sine right parenthesis super prime times left parenthesis z right parenthesis comma by the Chain Rule. So let f of z equals exp of sine of z times cosine of z, cap f of z equals exp of sine of z and script cap r equals double-struck cap c. Then f is continuous on script cap r, and cap f is a primitive of f on script cap r. Thus, by the Fundamental Theorem of Calculus,

    multiline equation row 1 integral over normal cap gamma e super sine of z times cosine of z times d times z equals cap f times left parenthesis pi solidus two right parenthesis minus cap f of zero row 2 Blank equals exp of sine of pi solidus two minus exp of sine of zero row 3 Blank equals e minus one full stop

    Remark: If you have a good deal of experience at differentiating and integrating real and complex functions, then you may have chosen to write down the primitive cap f of z equals e super sine of z of f of z equals e super sine of z times cosine of z straight away.

  • e.The integrand sine of z solidus cosine squared of z can be written as

    negative one divided by cosine squared of z times cosine super prime of z comma

    which equals

    left parenthesis h ring operator cosine right parenthesis super prime times left parenthesis z right parenthesis comma where h of z equals one solidus z full stop

    So let

    multiline equation row 1 Blank f of z equals sine of z solidus cosine squared of z comma row 2 Blank equation sequence part 1 cap f of z equals part 2 h of cosine of z equals part 3 one solidus cosine of z comma row 3 Blank script cap r equals double-struck cap c minus left parenthesis n plus one divided by two right parenthesis times pi colon n element of double-struck cap z full stop

    Then f is continuous on script cap r, and cap f is a primitive of f on script cap r. Thus, by the Fundamental Theorem of Calculus,

    multiline equation row 1 integral over normal cap gamma sine of z divided by cosine squared of z d z equals cap f of pi minus cap f of zero row 2 Blank equals one divided by cosine of pi minus one divided by cosine of zero row 3 Blank equation sequence part 1 equals part 2 negative one minus one equals part 3 negative two full stop

    (In this solution, note that the region script cap r does not contain the point pi solidus two, as cosine of pi solidus two equals zero; thus normal cap gamma cannot be chosen to be a path that contains pi solidus two. In particular, the real integral integral over zero under pi sine of x divided by cosine squared of x d x does not exist.)

Next we give a version of Integration by Parts for contour integrals.

Theorem 9 Integration by Parts

Let f and g be functions that are analytic on a region script cap r, and suppose that f super prime and g super prime are continuous on script cap r. Let normal cap gamma be a contour in script cap r with initial point alpha and final point beta. Then

integral over normal cap gamma f of z times g super prime of z d z equals left square bracket f of z times g of z right square bracket sub alpha super beta minus integral over normal cap gamma f super prime of z times g of z d z full stop

Proof

Let cap h of z equals f of z times g of z and h of z equals f super prime of z times g of z plus f of z times g super prime of z. Then h is continuous on script cap r, by hypothesis. Also, h has primitive cap h, since cap h is analytic on script cap r and
cap h super prime of z equals h of z comma

by the Product Rule for differentiation. It follows from the Fundamental Theorem of Calculus that

integral over normal cap gamma h of z d z equals left square bracket cap h of z right square bracket sub alpha super beta semicolon

that is,

integral over normal cap gamma left parenthesis f super prime of z times g of z plus f of z times g super prime of z right parenthesis d z equals left square bracket f of z times g of z right square bracket sub alpha super beta full stop

Using the Sum Rule (Theorem 5(a)) and rearranging the resulting equation, we obtain

integral over normal cap gamma f of z times g super prime of z d z equals left square bracket f of z times g of z right square bracket sub alpha super beta minus integral over normal cap gamma f super prime of z times g of z d z comma
as required.

Example 7

Use Integration by Parts to evaluate

integral over normal cap gamma z times e super two times z d z comma

where normal cap gamma is any contour from zero to pi times i.

Solution

We take f of z equals z, g of z equals one divided by two times e super two times z and script cap r equals double-struck cap c. Then f and g are analytic on script cap r, and f super prime of z equals one and g super prime of z equals e super two times z are continuous on script cap r.

Integrating by parts, we obtain

multiline equation row 1 integral over normal cap gamma z times e super two times z d z equals left square bracket z multiplication one divided by two times e super two times z right square bracket sub zero super pi times i minus integral over normal cap gamma one multiplication one divided by two times e super two times z d z row 2 Blank equals left parenthesis pi times i multiplication one divided by two times e super two times pi times i minus zero right parenthesis minus left square bracket one divided by four times e super two times z right square bracket sub zero super pi times i row 3 Blank equals one divided by two times pi times i minus left parenthesis one divided by four minus one divided by four right parenthesis row 4 Blank equals one divided by two times pi times i full stop

Exercise 13

Use Integration by Parts to evaluate the following integrals.

  • a.integral over normal cap gamma z times hyperbolic cosine of z times d times z comma where normal cap gamma is any contour from 0 to pi times i.

  • b.integral over normal cap gamma Log of z times d times z, where normal cap gamma is any contour from 1 to i lying in the cut plane double-struck cap c minus x element of double-struck cap r colon x less than or equals zero.

(Hint: For part (b), take f of z equals Log of z and g of z equals z.)

Answer

  • a.We take f of z equals z, g of z equals hyperbolic sine of z and script cap r equals double-struck cap c.

    Then f and g are analytic on script cap r, and f super prime of z equals one and g super prime of z equals hyperbolic cosine of z are continuous on script cap r.

    Integrating by parts, we obtain

    multiline equation row 1 integral over normal cap gamma z times hyperbolic cosine of z times d times z equals left square bracket z times hyperbolic sine of z right square bracket sub zero super pi times i minus integral over normal cap gamma one multiplication hyperbolic sine of z times d times z row 2 Blank equals left parenthesis pi times i times hyperbolic sine of pi times i minus zero right parenthesis minus left square bracket hyperbolic cosine of z right square bracket sub zero super pi times i row 3 Blank equals pi times i multiplication i times sine of pi minus left parenthesis cosine of pi minus hyperbolic cosine of zero right parenthesis row 4 Blank equation sequence part 1 equals part 2 zero minus left parenthesis negative one minus one right parenthesis equals part 3 two full stop
  • b.We take f of z equals Log of z, g of z equals z and script cap r equals double-struck cap c minus x element of double-struck cap r colon x less than or equals zero. Then f and g are analytic on script cap r, and f super prime of z equals one solidus z and g super prime of z equals one are continuous on script cap r.

    Integrating by parts, we obtain

    multiline equation row 1 integral over normal cap gamma Log of z times d times z equals left square bracket z times Log of z right square bracket sub one super i minus integral over normal cap gamma one divided by z multiplication z d z Blank row 2 Blank equals i times Log of i minus Log of one minus left square bracket z right square bracket sub one super i Blank row 3 Blank equation sequence part 1 equals part 2 negative pi solidus two minus left parenthesis i minus one right parenthesis equals part 3 left parenthesis one minus pi solidus two right parenthesis minus i full stop Blank

The Fundamental Theorem of Calculus is a useful tool when the function f being integrated has an easily determined primitive cap f. However, if the function f has no primitive, or if we are unable to find one, then we have to resort to the definition of an integral and use parametrisation. For example, we cannot use the Fundamental Theorem of Calculus to evaluate

integral over normal cap gamma z macron d z

along any contour, since the function f of z equals z macron has no primitive on any region.

To see why this is so, suppose that f is a function that is defined on a region in the complex plane. We observe that if fis not differentiable, then fhas no primitive cap f. This is because any differentiable complex function can be differentiated as many times as we like. Thus, if f has a primitive cap f, then cap f is differentiable with cap f super prime equals f. Hence f is also differentiable.

It follows that we cannot use the Fundamental Theorem of Calculus to evaluate integrals of non-differentiable functions such as

multiline equation row 1 Blank z long right arrow from bar z macron comma z long right arrow from bar Re of z comma z long right arrow from bar Im of z and z long right arrow from bar absolute value of z full stop

We conclude this section by proving the Fundamental Theorem of Calculus.

Proof

The proof of the Fundamental Theorem of Calculus is in two parts. We first prove the result in the case when normal cap gamma is a smooth path, and then extend the proof to contours.
  • a.Let normal cap gamma colon gamma of t left parenthesis t element of left square bracket a comma b right square bracket right parenthesis be a smooth path. Then

    multiline equation row 1 integral over normal cap gamma f of z d z equals integral over a under b f of gamma of t times gamma times super prime times left parenthesis t right parenthesis d t row 2 Blank equals integral over a under b cap f super prime of gamma of t times gamma times super prime times left parenthesis t right parenthesis d t row 3 Blank equals integral over a under b left parenthesis cap f ring operator gamma right parenthesis super prime times left parenthesis t right parenthesis d t comma

    by the Chain Rule. Now, if we write left parenthesis cap f ring operator gamma right parenthesis times left parenthesis t right parenthesis as a sum of its real and imaginary parts u of t plus i times v of t, then

    integral over a under b left parenthesis cap f ring operator gamma right parenthesis super prime times left parenthesis t right parenthesis d t equals integral over a under b u super prime of t d t plus i times integral over a under b v super prime of t d t full stop

    The Fundamental Theorem of Calculus for real integrals (Theorem 2) tells us that

    integral over a under b u super prime of t d t equals u of b minus u of a and integral over a under b v super prime of t d t equals v of b minus v of a full stop

    Hence

    equation sequence part 1 integral over normal cap gamma f of z d z equals part 2 left parenthesis u of b minus u of a right parenthesis plus i times left parenthesis v of b minus v of a right parenthesis equals part 3 cap f of beta minus cap f of alpha comma