Sequences, Series, and Special Functions 527- f(z, t) is a function of the complex variable z on the open set A, and
of the real variable t on R+.
2. lf(z,t)I::; M (a constant) for z EA and t ER+.
3. For every fixed z the function f is continuous with respect to t, and
for every fixed t it is analytic in A.- The integral J 000 f(z, t) dt is convergent for every z E A, and uniformly
convergent on every compact subset of A.
Then the function
is analytic in A, and
F(z)= 1
00f(z,t)dt
00 k
F(k)(z) = J 8 J(z, t) dt
8zk
0(8.2-1)(8.2-2)for every z E A, the integral on the right of (8.2-2) being uniformly
convergent on every compact subset of A.
Proof Consider any sequence of positive numbers {tn}~ such that tn <
tn+l and tn ---+ oo as n ---t oo, and define the function
Fn(z) = ltn J(z, t) dt
By Theorem 7.37 the functions Fn(z) are analytic in A, and Fn(z)---+ F(z)
for every z E A, the convergence being uniform on compact subsets of A
(by assumption 4). Hence, by Theorem 8.2, F(z) is analytic in A and
F~k)(z)---+ F(k)(z) for every z EA, with uniform convergence on compact
subsets of A. But in view of Theorem 7.37, we have
F(k)(z) = tn 8k f(z, t) dt
n Jo 8zkand it follows that
F(k)(z) =loo 8k~~:,t) dt
since { tn} ---t oo was an arbitrary sequence of increasing positive number.
Next we wish to apply Theorem 8.4 to prove that if f(z, t) satisfies in a
simply connected region G the four conditions enumerated in the statement
of that theorem, and C is a regular contour with graph contained in G, then
I [f f((,t)dt] d( ~ f [/ f(C,t)dC] dt (8.2-3)