vanhees71 said:
Hm, the asymptotics of Bessel functions can be found in many books. A very good one, introducing the various special functions needed in theoretical physics using physics motivations is A. Sommerfeld, Lectures on Theoretical Physics, vol. 6.
I believe I know what the asymptotic functions are, but my issue is how to use them to justify the inner coefficient and integrand in 8.92. Maybe I should post some of the details I had typed out.
I don't know why the latex is not working now, I am trying to fix it
So, if I simplify the full integrand expression in 8.90, I get, for the upper limit:
$$x \frac {I_m(x)}{ K_m(x)} K'(x\varrho/R) K(x\varrho/R) = - \frac{R}{2\varrho}exp(-2x(\varrho/R - 1)$$
and probably it is okay to neglect the -1 in the exponential here.
For the ##m \neq 0## lower limit terms,
$$x \frac {I_m(x)}{ K_m(x)} K'(x\varrho/R) K(x\varrho/R) = (\frac{R}{\varrho})^{2m+1}$$
which clearly will be small near the lower limit.
For the ##m = 0## lower limit term, I get:
$$x \frac {I_m(x)}{ K_m(x)} K'(x\varrho/R) K(x\varrho/R) = -\frac{R}{\varrho} -\frac{R}{\varrho}ln(\frac{R}{\varrho})[\frac{1}{(ln(2)-ln(x)}]$$
Where the first term should be small enough to neglect and probably the ##ln(2)## can be neglected as well.
So, for the upper limit I have:
$$-\frac{R}{2\varrho}exp(-2x(\varrho/R))$$
and for the lower limit I have:
$$\frac{R}{\varrho}ln(\frac{R}{\varrho})[\frac{1}{ln(x)}]$$
But I do not see how to actually get 8.92. How are these two results being combined?
Why would I multiply these functions but not their full coefficients? I can't make sense of the inner coefficient in 8.92. Why is it just the coefficient of 8.91? At a minimum why is there still a factor of pi in the numerator? The factor of pi from 8.91 is canceled by the one coming from the ##I_m(x)/K_m(x)## when simplifying the upper limit functions. And why does the coefficient of the lower limit function just getting ignored.
