Hi!

I have a couple of problems from the book: Mathematics for physics and physicists by W. Appel, which I can not solve.

The problems are:

1) Show that

$\displaystyle \int_1^x dt \frac{| \sin t|}{t} \underset{x\rightarrow \infty}{\sim} \frac{2}{\pi} \log(x) $

and

2) Show that

$\displaystyle \int_1^\infty dt \frac{e^{-xt}}{\sqrt{1+t^2}} \underset{x \rightarrow 0^+}{\sim} - \log(x) $.

Related to this is given first the defintion:

$\displaystyle f \underset{b}{\sim} g \Leftrightarrow \forall \epsilon>0, \exists \delta>0 \textrm{ s.t. } |x - b|<\delta \Rightarrow |f(x) - g(x)| < \epsilon |g(x)| $ ,

which in a slightly less general way, but perhaps more comprehensive way can be written as

$\displaystyle

\frac{f(x)}{g(x)} \underset{x\rightarrow b}{\rightarrow} 1 .

$

A theorem is also stated which says:

Let $\displaystyle g$ be a non-negative measurable function on $\displaystyle [ a, b [ $, and let $\displaystyle f$ be a measurable function on $\displaystyle [ a, b [ $. Assume that $\displaystyle g$ is not integrable and that $\displaystyle f$ and $\displaystyle g$ are both integrable on any interval $\displaystyle [ a, c ] $ where $\displaystyle c < b $. Then the asymptotic comparison between the functions extend to the integrals in the way

$\displaystyle f \underset{b}{\sim} g \Rightarrow \int_a^x f \underset{x \rightarrow b}{\sim} \int_a^x g $

and

$\displaystyle f \underset{b}{=} o(g) \Rightarrow \int_a^x f = \underset{x\rightarrow b}{o} \left( \int_a^x g \right).

$

My attempt at a solution:

I have tried to solve 1) by noting that

$\displaystyle

\frac{2}{\pi} \log (x) = \int_1^x dt \frac{2}{\pi t}

$.

After writing it like this I should be able to utilize the first part of the theorem.

The next step would then be to show that $\displaystyle \frac{|\sin x|}{x} \underset{x \rightarrow \infty}{\sim} \frac{2}{\pi x} $. However, this does not seem to hold as $\displaystyle \frac{|\sin x|}{x} \left(\frac{2}{\pi x}\right)^{-1} = \frac{\pi |\sin x|}{2}$ and this does not tend to 1 as $\displaystyle x \rightarrow \infty$ so I cannot conclude that the integrands are asymptotically equivalent.

For problem 2) I have even more problems. In this case the x-dependence is not in the limits of the integral and the theorem cannot be applied directly. I have attempted to rewrite the integral by the use of different variable substitutions, but I cannot obtain the required form.

I have worked quite extensively on both of these problems, but I cannot seem to solve them. They are also the only two problems related to the theorem in the chapter I am studying. Either I do not fully comprehend the theorem, or there is something missing. I am a bit unsure about the definition for equivalence between two functions $\displaystyle f \underset{b}{\sim} g $ since, in the book, the exact meaning of this is only properly defined for sequences. However, the definition I give above is the direct generalization of this and should be correct.

I am grateful for any suggestions, or help I can get.