113

Chapter 5

Contour Integration

An introductory calculus course introduces the concepts of differentiation and integration

associated with functions of a real variable. Indefinite integration of a real function was

defined as the inverse operation of differentiation of the real function. A definite integral

was defined as the limit of a summation process representing area under a curve. In complex

variable theory one can have indefinite and definite integrals but their physical interpretation

is not quite the same as when dealing with real quantities. In addition to indefinite and

definite integrations one must know properties of contour integrals.

Contour integration

Let C denote a curve in the z-plane connecting two points z = a and z = b as illustrated

in figure 5-1.

Figure 5-1. Curve in the z-plane.

The curve C is assumed to be a smooth curve represented by a set of parametric equations

x = x(t), y = y(t), ta ≤ t ≤ tb

The equation z = z(t) = x(t) + iy(t), for ta ≤ t ≤ tb represents points on the curve C with the

end points given by

z(ta) = x(ta) + iy(ta) = a and z(tb) = x(tb) + iy(tb) = b.

Divide the interval (ta, tb) into n parts by defining a step size h =tb − ta

nand letting

t0 = ta, t1 = ta + h, t2 = ta + 2h, . . . , tn = ta + nh = ta + n(tb − ta)

n= tb. Each of the values

ti, i = 0, 1, 2, . . . , n, gives a point zi = z(ti) on the curve C. For f(z), a continuous function at all

points z on the curve C, let ∆zi = zi+1 − zi and form the sum

114

Sn =n−1∑

i=0

f(ξi)∆zi =n−1∑

i=0

f(ξi)(zi+1 − zi), (5.1)

where ξi is an arbitrary point on the curve C between the points zi and zi+1. Now let n increase

without bound, while |∆zi| approaches zero. The limit of the summation in equation (5.1) is

called the complex line integral of f(z) along the curve C and is denoted

∫

C

f(z) dz = limn→∞

n−1∑

i=0

f(ξi)∆zi. (5.2)

If f(z) = u(x, y) + iv(x, y) is a function of a complex variable, then we can express the

complex line integral of f(z) along a curve C in the form of a real line integral by writing∫

C

f(z) dz =∫

C

[u(x, y) + iv(x, y)] (dx + idy)

=∫

C

[u(x, y) dx− v(x, y) dy] + i

∫

C

[v(x, y) dx + u(x, y) dy](5.3)

where x = x(t), y = y(t), dx = x′(t) dt and dy = y′(t) dt are substituted for the x, y, dx and dy

values and the limits of integration on the parameter t go from ta to tb. This gives the integral

∫

C

f(z) dz =∫ tb

ta

[u(x(t), y(t)) x′(t) − v(x(t), y(t))y′(t)] dt + i

∫ tb

ta

[v(x(t), y(t)) x′(t) + u(x(t), y(t)) y′(t)] dt

Now both the real part and imaginary parts are evaluated just like the real integrals you

studied in calculus.

If the parametric equations defining the curve C are not given, then you must construct

the parametric equations defining the contour C over which the integration occurs. Complex

line integrals along a curve C involve a summation process where values of the function being

integrated must be known on a specified path C connecting points a and b. We will find that

in special cases the value of the complex integral is very much dependent upon the path of

integration while in other special circumstances the value of the line integral is independent

of the path of integration joining the end points. In some special circumstances the path

of integration C can be continuously deformed into other paths C∗ without changing the

value of the complex integral. In this chapter we present various theorems associated with

integration involving analytic functions f(z) which are well defined over specific regions of

the z-plane.

The integration of a function along a curve is called a line integral. A familiar line inte-

gral is the calculation of arc length between two points on a curve. Let ds2 = dx2 +dy2 denote

an element of arc length squared and let C denote a curve defined by the parametric equa-

tions x = x(t), y = y(t), for ta ≤ t ≤ tb, then the arc length L between two points a = [x(ta), y(ta)]

and b = [x(tb), y(tb)] on the curve is given by the integral

115

L =∫

C

ds =∫ tb

ta

√(dx

dt

)2

+(

dy

dt

)2

dt

If the curve C represents a wire with variable density f(x, y) [gm/cm], then the total mass

m of the wire between the points a and b is given by m =∫

C

f(x, y) ds which can be thought

of as the limit of a summation process. If the curve C is partitioned into n pieces of lengths

∆s1,∆s2, . . . , ∆si, . . ., then in the limit as n increases without bound and ∆si approaches zero,

one can express the total mass m of the wire as the limiting process

m =∫

C

f(x, y) ds = lim∆si→0n→∞

n∑

i=1

f(x∗i , y∗i ) ∆si =

∫ tb

ta

f(x(t), y(t))

√(dx

dt

)2

+(

dy

dt

)2

dt

where (x∗i , y

∗i ) is a general point on the ∆si arc length.

Whenever the values of x and y are restricted to lie on a given curve defined by x = x(t)

and y = y(t) for ta ≤ t ≤ tb, then integrals of the form

I =∫

C

P (x, y) dx + Q(x, y) dy =∫ tb

ta

P (x(t), y(t))x′(t) dt + Q(x(t), y(t))y′(t) dt (5.4)

are called line integrals and are defined by a limiting process such as above. Line integrals

are reduced to ordinary integrals by substituting the parametric values x = x(t) and y = y(t)

associated with the curve C and integrating with respect to the parameter t. The above line

integral is sometimes written in the form∫

C

f(z) dz =∫ tb

ta

f(z(t)) z′(t) dt (5.5)

where z = z(t) is a parametric representation of the curve C over the range ta ≤ t ≤ tb.

Whenever the curve C is not a smooth curve, but is composed of a finite number of arcs

which are smooth, then the curve C is called piecewise smooth. If C1, C2, . . . , Cm denote the

finite number of arcs over which the curve is smooth and C = C1 ∪C2 ∪ · · ·∪Cm, then the line

integral can be broken up and written as a summation of the line integrals over each section

of the curve which is smooth and one would express this by writing∫

C

f(z) dz =∫

C1

f(z) dz +∫

C2

f(z) dz + · · ·+∫

Cm

f(z) dz (5.6)

Indefinite integrationIf F (z) is a function of a complex variable such that

dF (z)dz

= F ′(z) = f(z), then F (z) is called

an anti-derivative of f(z) or an indefinite integral of f(z). The indefinite integral is denoted

using the notation∫

f(z) dz = F (z) + c where c is a constant and F ′(z) = f(z) Note that the

addition of a constant is included because the derivative of a constant is zero. Consequently,

any two functions which differ by a constant will have the same derivatives.

116Example 5-1. (Indefinite integration)

Let F (z) = 3 sin z + z3 + 5z2 − z withdF

dz= F ′(z) = 3 cos z + 3z2 + 10z − 1, then one can write

∫(3 cos z +3z2+10z−1) dz = 3 sin z +z3+5z2−z +c where c is an arbitrary constant of integration

The table 5.1 gives a short table of indefinite integrals associated with selected functions

of a complex variable. Note that the results are identical with those derived in a standard

calculus course.

Table 5.1 Short Table of Integrals

1.∫

zn dz =zn+1

n + 1+ c, n 6= −1 11.

∫sinh z dz = cosh z + c

2.∫

dz

z= log z + c 12.

∫cosh z dz = sinh z + c

3.∫

ez dz = ez + c 13.∫

tanh z dz = log (cosh z) + c

4.∫

kz dz =kz

log k+ c, k is a constant 14.

∫sech2z dz = tanh z + c

5.∫

sin z dz = − cos z + c 15.∫

dz√z2+α2 = log (z +

√z2 + α2) + c

6.∫

cos z dz = sin z + c 16.∫

dzz2+α2 = 1

α tan−1 zα + c

7.∫

tan z dz = log sec z + c = − log cos z + c 17.∫

dzz2−α2 = 1

2αlog

(z−αz+α

)+ c

8.∫

sec2 z dz = tan z + c 18.∫

dz√α2−z2 = sin−1 z

α+ c

9.∫

sec z tan z dz = sec z + c 19.∫

eαz sin βz dz = eαz α sin βz − β cos βzα2 + β2 + c

10.∫

csc z cot z dz = −csc z + c 20.∫

eαz cos βz dz = eαz α cos βz + β sin βzα2 + β2 + c

c denotes an arbitrary constant of integration

Definite integrals

The definite integral of a complex function f(t) = u(t) + iv(t) which is continuous for

ta ≤ t ≤ tb has the form ∫ tb

ta

f(t) dt =∫ tb

ta

u(t) dt + i

∫ tb

ta

v(t) dt (5.7)

and has the following properties.

1. The integral of a linear combination of functions is a linear combination of the integrals

of the functions or

∫ tb

ta

[c1f(t) + c2g(t)] dt = c1

∫ tb

ta

f(t) dt + c2

∫ tb

ta

g(t) dt

where c1 and c2 are complex constants.

1172. If f(t) is continuous and ta < tc < tb, then

∫ tb

ta

f(t) dt =∫ tc

ta

f(t) dt +∫ tb

tc

f(t) dt

3. The modulus of the integral is less than or equal to the integral of the modulus∣∣∣∣∫ tb

ta

f(t) dt

∣∣∣∣ ≤∫ tb

ta

|f(t)| dt

4. If F = F (t) is such thatdF

dt= F ′(t) = f(t) for ta ≤ t ≤ tb, then

∫ tb

ta

f(t) dt = F (t)]tb

ta= F (tb) − F (ta)

5. If G(t) is defined G(t) =∫ t

ta

f(t) dt, thendG

dt= G′(t) = f(t)

6. The conjugate of the integral is equal to the integral of the conjugate∫ tb

ta

f(t) dt =∫ tb

ta

f(t) dt

7. Let f(t, τ ) denote a function of the two variables t and τ which is defined and con-

tinuous everywhere over the rectangular region R = {(t, τ ) | ta ≤ t ≤ tb, τc ≤ τ ≤ τd}. If

g(τ ) =∫ tb

ta

f(t, τ ) dt and the partial derivatives of f exist and are continuous on R, then

dg

dτ=

∫ tb

ta

∂f(t, τ )∂τ

dt

which shows that differentiation under the integral sign is permissible.

Assume F (z) is an analytic function with derivative f(z) = dFdz

and z = z(t) for t1 ≤ t ≤ t2

is a piecewise smooth arc C in a region R of the z-plane, then one can write∫

C

f(z) dz =∫ t1

t1

dF

dzdz = F (z(t))

t2

t1

= F (z(t2)) − F (z(t1))

This is a fundamental integration property in the z-plane. Note that if F (z) = U + iV is

analytic and f(z) = u + iv, then one can write

F ′(z) =∂U

∂x+ i

∂V

∂x= u + iv =

∂V

∂y− i

∂U

∂y

and consequently,∫

C

f(z) dz =∫

C

(u + iv)(dx + idy)

=∫

C

(u dx− v dy) + i

∫

C

(v dx + u dy)

=∫ t2

t1

∂U

∂xx′(t) dt +

∂U

∂yy′(t) dt + i

∫ t2

t1

∂V

∂xx′(t) dt +

∂V

∂yy′(t) dt

=∫ z(t2)

z(t1)

dU + i

∫ z(t2)

z(t1)

dV =∫ z(t2)

z(t1)

dF = F (z)z(t2)

z(t1)

= F (z(t2)) − F (z(t1))