in equation (12.183) is called the complex line integral of f(z)along the curve Cand
is denoted ∫
C
f(z)dz = limn→∞
n∑− 1
i=0
f(ξi)∆ zi. (12.184)
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 Cin 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 ]
(12.185)
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 tgo from tato 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 of real variables.
If the parametric equations defining the curve Care not given, then you must
construct the parametric equations defining the contour C over which the integra-
tion 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 aand b. 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 Ccan be continuously
deformed into other paths C∗ without changing the value of the complex integral.
If you take a course in complex variable theory you will be introduced to various
theorems and properties 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 integral is the calculation of arc length between two points on a curve. Let
ds^2 =dx^2 +dy^2 denote an element of arc length squared and let C denote a curve
defined by the parametric equations x=x(t), y =y(t),for ta≤t≤tb, then the arc
