Re: integral(0,inf){ t k exp(-kt) }dt

On Sun, 9 Aug 2009, David C. Ullrich wrote:

(i) Suppose that F(x,t) and the partial F_x(x,t) are
continuous in the compact rectangle [a,b] x [c,d].
Define

f(x) = int_c^d F(x,t) dt (x in [a,b]).

Then f is differentiable on (a,b), with

f'(x) = int_c^d F_x(x,t) dt.

That's not enough for the current problem because
we're taking an integral from 0 to infinity. But here's
something else from calculus:

(ii) If f_n is differentiable, f_n -> f pointwise and
f_n' -> g uniformly then f is differentiable and f'=g.

Now let f_n = the integral from 1/n to n in the original
problem. (i) shows that f_n' is what it should be and then
you can use (ii) to show that f' is what it should be.

Does this mean that (i) can be extended to

(ii) Suppose that F(x,t) and the partial F_x(x,t) are
continuous in the sigma compact rectangle [a,b] x [c,oo)
Define

f(x) = int_c^d F(x,t) dt (x in [a,b]).

Then f is differentiable on (a,b), with

f'(x) = int_c^oo F_x(x,t) dt.

.

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