MTH 312 - Final

Definitons, Theorems, & Lemma's For 8.4

Definition 8.4.2.

Given t > 0, define the exponential function t^x to be

t^x = e^{x log t} for all x ∈ R.

Definition 8.4.3.

Assume f is defined on [a, ∞) and integrable on every interval of the form [a, b]. Then define ∫^∞_a f to be

lim _b→∞ ∫^b_a f,

provided the limit exists. In this case we say the improper integral ∫^∞_a f converges.

Definition 8.4.4.

A function f : D → R is continuous at (x₀, t₀) if for all ε > 0, there exists δ > 0 such that whenever ||(x, t) − (x₀, t₀)|| < δ, it follows that

|f(x, t) − f(x₀, t₀)| < ε.

Theorem 8.4.5.

If f(x, t) is continuous on D, then F(x) = ∫^d_c f(x, t)dt is uniformly continuous on [a, b].

Theorem 8.4.6.

If f(x, t) and f_x(x, t) are continuous on D, then the function F(x) = ∫^d_c f(x, t)dt is differentiable and

F’(x) = ∫^d_c f_x(x, t)dt.

Definition 8.4.7.

Given f(x, t) defined on D = {(x, t) : x ∈ A, c ≤ t}, assume F(x) = ∫^∞_c f(x, t)dt exists for all x ∈ A. We say the improper integral converges uniformly to F(x) on A if for all ε > 0, there exists M > c such that

|F(x) − ∫^d_c f(x, t)dt| < ε

for all d ≥ M and all x ∈ A.

Theorem 8.4.8.

If f(x, t) is continuous on D = {(x, t) : a ≤ x ≤ b, c ≤ t}, then

F(x) = ∫^∞_c f(x, t)dt

is uniformly continuous on [a, b], provided the integral converges uniformly.

Theorem 8.4.9.

Assume the function f(x, t) is continuous on D = {(x, t) : a ≤ x ≤ b, c ≤ t} and F(x) = ∫^∞_c f(x, t)dt exists for each x ∈ [a, b]. If the derivative function f_x(x, t) exists and is continuous, then

(7) F’(x) = ∫^∞_c f_x(x, t)dt,

provided the integral in (7) converges uniformly.

Definition 8.4.10

For x ≥ 0, define the factorial function

x! = ∫^∞_c t^xe^−t dt

Theorem 8.4.11 (Bohr–Mollerup Theorem).

There is a unique positive function f defined on x ≥ 0 satisfying

(i) f(0) = 1
(ii) f(x + 1) = (x + 1)f(x), and
(iii) log(f(x)) is convex.

Because x! satisfies properties (i), (ii), and (iii), it follows that f(x) = x!.