1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 501

Now choose h such that ihl < 1/ 2 r. Then iu -zl =rand iu -z - hi>^1 / 2 r.

From (7.27-5) we have

F(z+h) = ~ jdej f(u,()du

27ri u - z - h

c 'Y

so that

F(z + h)-F(z) = _1 jdej f(u, ()du

h 27fi (u-z)(u-z-h)

c 'Y

= _1 J def f(u,()du + _l J def hf(u,e)du

27fi (u - z)^2 27fi (u -z)^2 (u - z - h)

c 'Y c 'Y

(7.27-6)

since

~jdej. hf(u,()du 1 lhlM

27fi (u - z)^2 (u - z - h) ~ 27f %ra (^2 7rr)L(C)

c 'Y

= 2ihlM L(C)

r2

the last term in. (7.27-6) tends to zero as h -t 0. Hence the left-hand side

of this equation has a limit as h -t 0, and we get

F'(z) = ~ J def f(u,()du = J ~ f(z, ()de

27fi (u -z)^2 az

c 'Y c

by taking (7.27-4) into account.

This result shows that under the_ stated assumptions, it is permissible

to differentiate the integral under the integral sign.

Example If F(z) = fc(e^2 z + 3z^3 e^2 ) d(, z E {z: lzl < R}, then F'(z) =
f c(2e^2 z + 9z^2 e^2 ) de for any z such that lzl < R.

7.28 Schwarz's and Poisson's Formulas

Cauchy's formula shows that the values of an analytic function on a region
bounded by one of several simple closed contours are determined by the
values assumed by the function on the total boundary of the region. This
is also true for the real part of an analytic function, and it can be shown
that up to an additive constant both the imaginary part and the analytic
function itself are determined in such a region by the values assumed by