6
1 Preliminaries
Theorem 1.2.3 (The exchangeability of derivative and integral order) If f (x, y) and
f y (x, y) are continuous in the rectangular D[a, b; c, d], then the integral ϕ(y) =
b
a f (x, y)dx is derivable in [c, d], and where is
d
dy
b
a
f (x, y)dx =
b
a
f y (x, y)dx
(1.2.4)
namely the order of integral and derivation can be exchanged, this property is called
the differentiating under the integral sign or differentiation under the integral
sign.
Proof Suppose that F(y) =
b
a f y (x, y)dx, since the function f y (x, y) is continuous
in D, so its double integral in D can be exchanged integral order, thus for any y in
the interval [c, d], there is
y
c
F(y)dy =
y
c
b
a
f y (x, y)dx
dy =
b
a
[
y
c
f y (x, y)dy]dx
=
b
a
[ f (x, y) − f (x, c)]dx = ϕ(y) − ϕ(c)
Because F(y) is continuous in the interval [c, d], taking derivative to the both
ends of the above expression, we obtain
ϕ
(y) = F(y) =
b
a
f y (x, y)dx
Quod erat demonstrandum.
Sometimes such a kind of integral with parameter could be met, its upper limt
and lower limit are also a function of the parameters, namely
ϕ(y) =
β(y)
α(y)
f (x, y)dx
(1.2.5)
There is the following theorem about it:
Theorem 1.2.4 Suppose that both the function f (x, y) and f y (x, y) are continuous in the closed rectangular domain D[a, b; c, d], the function α(y) and β(y) are
derivable in the interval [c, d], and when c ≤ y ≤ d, there are a ≤ α(y) ≤ b,
a ≤ β(y) ≤ b, then the function ϕ(y) =
β(y)
α(y) f (x, y)dx is continuous in the
interval [c, d], and there is
ϕ (y) =
d
dy
β(y)
α(y)
f (x, y)dx =
β(y)
α(y)
f y (x, y)dx + f (β(y), y)β (y) − f (α(y), y)α (y) (1.2.6)
Equation (1.2.6) is called the Leibniz formula.