1. Ch11.4~11.6 (Ch12.3e) Hankel Functions, Modified Bessel Functions, Asymptotic Expansions
講者： 許永昌 老師

2. Contents Contour Integral Representation of the Hankel Functions
Steepest descents  Asymptotic series
Modified Bessel function

3. Contour Integral Representation of the Hankel Functions (請預讀P707~P710)
Since
g(x,t) will have essential singularities at t=0, .
Residue theorem?
It is worked for n is an integer number case; otherwise, it will have a branch point at t=0.
Derived by yourself:

4. Contour Integral Representation of the Hankel Functions (continue)
You will find both t=0+ and t=- will let fg=0 Re{x}>0.
Bessel Jn :
Check:

5. Contour Integral Representation of the Hankel Functions (continue)

6. Contour Integral Representation of the Hankel Functions (continue)
Prove that
Prove that at first.
Use these equations to find out J-n(x).
Finally, we get the formula of Nn shown in Ch11.3.

7. Asymptotic series of the Hankel Functions (請預讀P719~P723，此page只大略講)
When z and Re{t-1/t}<0, Exp(z/2[t-1/t])0
i.e. (|t|-1/|t|)cos(ang(t)) <0
The main contribution is at the
saddle point i.
Steepest descent:  

8. Modified Bessel Functions, In & Kn (請預讀P713~P716，另一本無)
Helmholtz eq. : [2+k2]y=0
Bessel eq. : x2y’’+xy’+[x2-n2]y= (1)
Modified Helmholtz eq. : [2-k2]y=0
Modified Bessel eq. : x2y’’+xy’-[x2+n2]y= (2)
Eq. (2) can be transformed from Eq. (1) by the transformation x ix.
In(x)i-nJn(ix)=e-inp/2Jn(xeip/2). [i-n is used to make sure In(x) if x].
Kn(x)p/2 in+1H(1)n(ix). [in+1 is used to make sure Kn(x) if x. Besides, it will tend to zero when x.]

9. Homework
11.6.3(a~c) (12.3.2e)
(12.3.3e)
11.4.7