410 Chapter^71. Ref: f(t)dt = J:Ref(t)dt, Imf: f(t)dt = J:Imf(t)dt
- If f 1 and h are continuous on [a, b], then J:[f1(t) + f2(t)] dt =
1: fi(t) dt + 1: h(t) dt
3. If a < c < b, then J: f(t) dt = J: f(t) dt + J: f(t) dt
4. 1: J(t)dt = -Iba J(t)dt
- k J: f(t) dt = J: kf(t) dt, where k denotes a complex constant or,
more generally, a complex function independent of t.
6. 1: f(t) dt = 1: f(t) dt
- I 1: f(t) dtl ~ 1: lf(t)I dt
- Let t = g( r) be a real function of the real variable r with a continuous
derivative g'(r) inc~ r ~ d, and let f be continuous on g([c,d]), and
suppose that a= g(c), b = g(d). Then
lb f(t)dt = 1d f[g(r)]g'(r)dr
9. If F(t) is such that F'(t) = f(t) for a~ t::::; b, then
lb f(t) dt = F(b) -F(a)
(7.2-2)(7.2-3)10. Let G(t) = J: f(r)dr, a::::; t::::; b. Then G'(t) = f(t) for a::::; t::::; b.
...
Corollary 9.t' The function G(t) as defined above is continuous on [a, b].
It is to be understood that the derivative of G, as well as its continuity, at
a is from the right, while at b is from the left.- Let F(s) = J: f(t,s)dt, where f(t,s) is a continuous function in the
rectangle
If the partial derivative fs(t,s) exists and is continuous on R, then
the derivative F' ( s) exists for every s E [ c, d] and is given by
· F'(s) =lb fs(t,s)dt (7.2-4)
Corollary 7 .2 The function F( s) as defined above is continuous on [ c, d].
Proofs The equalities in 1 are only restatements of (7.2-1). Properties
2, 3, and 4 are easily proven by using the corresponding properties of real
integrals. Property 5 is obviously true for k real. To verify that the property