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

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

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:

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:

Contour Integral Representation of the Hankel Functions (continue)

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.

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:  

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=0 ----(1) Modified Helmholtz eq. : [2-k2]y=0 Modified Bessel eq. : x2y’’+xy’-[x2+n2]y=0 ----(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.]

Homework 11.6.3(a~c) (12.3.2e) 11.6.5 (12.3.3e) 11.4.7