Approximating integrals of Bessel functions

[pherytic]

I edited this to remove some details/attempts that I no longer think are correct or helpful.

But my core issue is I have never seen this approach to approximating integrals that is used in the attached textbook image. Any more details on what is happening here, or advice on where to learn more about these methods would be very appreciated.

 

[vanhees71]

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.

 

[pherytic]

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.