Real and Complex Analysis (Rudin)

376 REAL AND COMPLEX ANALYSIS
cl>,,(w) = - LOao!(X)e- WX dx (Re w < 0). (9)
cl>o and cl>" are holomorphic in the indicated half planes because of (2).
The significance of the functions cl>/X to (4) lies in the easily verified relation
L:.r.(X)e-itX dx = cl>o(€ + it) - cl>,,( -€ + it) (t real). (10)
Hence we have to prove that the right side of (10) tends to 0 as €-+ 0, if t > A
and ift < -A.
We shall do this by showing that any two of our functions cl>/X agree in
the intersection of their domains of definition, i.e., that they are analytic continuations
of each other. Once this is done, we can replace cl>o and cl>" by cl>,,/2
in (10) if t < -A, and by cl>-,,/2 if t> A, and it is then obvious that the
difference tends to 0 as € -+ o.
So suppose 0 < {J - ex < n. Put
ex+{J
Y=-2-'
{J-ex
" = cos -2- > o.
(11)
so that w E nIX n np as soon as I wi> AI". Consider the integral
l!(z)e- W % dz
over the circular arc r given by r(t) = reit, ex ::; t ::; {J. Since
Re (- wz) = - I wi r cos (t - y)::; - I w I r",
the absolute value of the integrand in (13) does not exceed
C exp {(A - I wi ,,)r}.
(12)
(13)
(14)
If I wi> AI" it follows that (13) tends to 0 as r-+ 00.
We now apply the Cauchy theorem. The integral of !(z)e- W % over the
interval [0, reiP] is equal to the sum of (13) and the integral over [0, rei/X].
Since (13) tends to 0 as r-+ 00, we conclude that cl>/X(w) = cl>p(w) if w = I w I e- iy
and I wi> AI", and then Theorem 10.18 shows that cl>/X
and cl>p coincide in
the intersection of the half planes in which they were originally defined.
This completes the proof.
19.4 Remarks Each of the two preceding proofs depended on a typical application
of Cauchy's theorem. In Theorem 19.2 we replaced integration over
one horizontal line by integration over another to show that 19.2(15) was
I I I I