1550251515-Classical_Complex_Analysis__Gonzalez_

688 Chapter 9

we look for a meromorphic (in particular, holomorphic) function F(z) in
C, or at least on either Imz;:::: 0 or Imz :S 0 (sometimes for a meromorphic
branch of a multiple-valued function) such that either F(x) = f(x), or
ReF(z) == f(x), or ImF(z) = J(x) on the appropriate infinite interval of
the real axis, except possibly at a finite number of points where F(z) may
have poles or branch points. This function F(z) may or may not exist, and
the method is applicable only if such a function can be found. However,

in a large number of cases of interest f ( x) is composed of algebraic and/ or

elementary transcendental functions all of which admit extensions to the

complex domain, and a F(z) with the stated properties can then be easily

produced. As customary, in what follows we use the same symbol f to

denote a real function as well as its extension to the complex domain. Yet

there are some cases in which we start with a complex f(z) and seek to

evaluate an improper integral of its restriction f ( x) to the real line.

For instance, if f(x) = (sinx)/x we may use F(z) = eiz/z since F(x) =
eix /x = (cosx + i sinx)/x (x-/:-0), and J(x) = Imeix /x.
Similarly, if J(x) = lnx/(x^2 + 4) (x > 0), we may choose f(z) =
logz/(z^2 + 4) with a branch of logz specified by logz = lnlzl + iB,

-B 0 < B :S 27r - 80 , with 0 < Bo :S 7r, so that for z = x > 0, log z = ln x.

Once the choice of f(z) has been made the method consists in evaluating

fc+ J(z) dz by the residue theorem, where the contour c+ consists of the

real interval [-R, R] (or [ 0, R]) and either a half-circle passing through the

points (--R, 0) and (R, 0), or, an arc of a circle passing through (R, 0) and

Reiot, with center at the origin. Alternatively, we may use a portion of a

rectangle, or of other simple geometric figure, joining the points (-R, 0)

and (R, 0) (Fig. 9.12). If we let I' = C - [-R, R], the method succeeds if

lira j f(z)dz

R-->oo

r+

exists. In case f(z) has either poles or branch points on the real axis such

points are bypassed by means of indentations made of small half-circles (or

r r

R -R 0 R -R 0 R

Fig. 9.12