From: Stephen Chinn Subject: Re: Modified Bessel Function: a name Date: Wed, 29 Dec 1999 07:10:22 -0500 Newsgroups: sci.math.num-analysis Pawel F. Gora wrote: > During some calculations I came across the function K_n(z), a > solution to the modified Bessel equation. snip... > > > I am a bit confused: What is then the name of this function? > Or perhaps, What name of this function is now most commonly used? > > What I am really interested in is not the name, though, but the > asymptotics. How does K_n(z) behave for n real, z real, positive > and small (z --> 0+)? The function appears to have a pole at zero, > but does it diverge as z^{-s}? Or logarithmically? Or in a more > complicated manner? Bateman gives formulae for large |z|, and > so do Gradshteyn and Ryzhik; I can't find any sources for the > case I am interested in. Any references (on-line or published > work) will be greatly appreciated. > > All the best, > > Pawel Gora > Institute of Physics, Jagellonian University, Cracow, Poland > A physical entity does not do what it does because it is what it is, > but is what it is because it does what it does. See Abramowitz & Stegun, "Handbook of Mathematical Functions.." They call it a modified Bessel function. The limit for small argument z ->0 is K(sub nu)(z) ~ (1/2) Gamma(nu) ( z/2)^(-nu) ============================================================================== From: harper@mcs.vuw.ac.nz (John Harper) Subject: Re: Modified Bessel Function: a name Date: 29 Dec 1999 21:25:12 GMT Newsgroups: sci.math.num-analysis In article <3869FA2D.DA9A9416@ll.mit.edu>, Stephen Chinn wrote: > >See Abramowitz & Stegun, "Handbook of Mathematical Functions.." >They call it a modified Bessel function. > >The limit for small argument z ->0 is >K(sub nu)(z) ~ (1/2) Gamma(nu) ( z/2)^(-nu) The above is only for real part of nu > 0. If you want nu < 0 use K(sub nu)(z) = K(sub -nu)(z). If nu = 0, RHS must be replaced by -ln(z). The Abramowitz+Stegun formulae are on p375 nos. 9.6.6, 9.6.8 and 9.6.9. John Harper, School of Mathematical and Computing Sciences, Victoria University, Wellington, New Zealand e-mail john.harper@vuw.ac.nz phone (+64)(4)463 5341 fax (+64)(4)463 5045 ============================================================================== From: wei-choon ng Subject: Re: Modified Bessel Function: a name Date: 28 Dec 1999 20:12:01 GMT Newsgroups: sci.math.num-analysis Pawel F. Gora wrote: > and Ryzhik (Tables of Integrals etc) call it modified Hankel's > function. [snip] > case I am interested in. Any references (on-line or published > work) will be greatly appreciated. Check out the Handbook of mathematical functions by Abromowitz and Stegun. It's a nicer function to use than the Hankel's function due to its exponentially decaying nature. Be careful though that the recurrence relations for this modified Hankel's functions are somewhat different from those of the normal Bessel and Hankel's functions. I got myself into a fix while evaluating the derivatives of the modified Hankel's functions because there was a mistake in one of the books I used. Tough luck! Wei-Choon. -- ============================================================================== [Here is Maple's terminology regarding this family of functions, edited from their help pages. --djr] Function: BesselI, BesselJ - The Bessel functions of the first kind Function: BesselK, BesselY - The Bessel functions of the second kind Function: HankelH1, HankelH2 - The Hankel functions (Bessel functions of the third kind) Function: StruveH, StruveL - The Struve functions Function: AngerJ - The Anger function Function: WeberE - The Weber function Function: KelvinBer, KelvinBei - The Kelvin functions ber and bei Function: KelvinKer, KelvinKei - The Kelvin functions ker and kei Calling Sequence: BesselI(v, x) [etc.] with the parameters v - an algebraic expression (the order or index) x - an algebraic expression (the argument) Descriptions: - BesselJ and BesselY are the Bessel functions of the first and second kinds, respectively. They satisfy Bessel's equation: 2 2 2 x y'' + x y' + (x - v ) y = 0 - BesselI and BesselK are the modified Bessel functions of the first and second kinds, respectively. They satisfy the modified Bessel equation: 2 2 2 x y'' + x y' - (x + v ) y = 0 - HankelH1 and HankelH2 are the Hankel functions, also known as the Bessel functions of the third kind. They also satisfy Bessel's equation, and are related to BesselJ and BesselY by HankelH1(v,x) = BesselJ(v,x) + I*BesselY(v,x) HankelH2(v,x) = BesselJ(v,x) - I*BesselY(v,x) - The Struve function StruveH(v,x) solves the inhomogeneous Bessel equation v + 1 2 2 2 4 (1/2 x) x y'' + x y' + (x - v ) y = -------------------- 1/2 Pi GAMMA(v + 1/2) - The modified Struve function StruveL(v,x) solves the inhomogeneous Bessel equation v + 1 2 2 2 (1/2 x) x y'' + x y' - (x + v ) y = 4 -------------------- 1/2 Pi GAMMA(v + 1/2) Reference: M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chapter 12. - The Anger function AngerJ(v,x) solves the inhomogeneous Bessel equation 2 2 2 (x - v) sin(v Pi) x y'' + x y' + (x - v ) y = ----------------- Pi - The Weber function WeberE(v,x) solves the inhomogeneous Bessel equation 2 2 2 (v - x) cos(v Pi) - (v + x) x y'' + x y' + (x - v ) y = --------------------------- Pi Reference: M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chapter 12, and G.N. Watston, A Treatise on the Theory of Bessel Functions, chapter 10. - The Kelvin functions (sometimes known as the Thompson functions) are defined by the following equations: KelvinBer(v,x) + I*KelvinBei(v,x) = BesselJ(v,x*exp(3*I*Pi/4)) KelvinBer(v,x) - I*KelvinBei(v,x) = BesselJ(v,x*exp(-3*I*Pi/4)) KelvinKer(v,x) + I*KelvinKei(v,x) = exp(-v*Pi*I/2)*BesselK(v,x*exp(I*Pi/4)) KelvinKer(v,x) - I*KelvinKei(v,x) = exp(-v*Pi*I/2)*BesselK(v,x*exp(-I*Pi/4)) KelvinHer(v,x) + I*KelvinHei(v,x) = HankelH1(v,x*exp(3*I*Pi/4)) KelvinHer(v,x) - I*KelvinHei(v,x) = HankelH2(v,x*exp(-3*I*Pi/4)) The Kelvin functions are all real valued for real x and positive v. Reference: M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, section 9.9, and A. Erdelyi, Higher Transcendental Functions, section 7.2.3 . Function: AiryAi, AiryBi - The Airy wave functions Calling Sequence: AiryAi(x) AiryBi(x) AiryAi(n,x) AiryBi(n,x) Parameters: x - an algebraic expression n - an algebraic expression, assumed to be a non-negative integer Description: - The Airy wave functions AiryAi and AiryBi are linearly independent solutions for w in the equation w''-x*w=0. Specifically, AiryAi(x) = c1*0F1( ; 2/3; x^3/9) - c2*x*0F1( ; 4/3; x^3/9) AiryBi(x) = 3^(1/2) * [c1*0F1( ; 2/3; x^3/9) + c2*x*0F1( ; 4/3; x^3/9)] where 0F1 is the generalized hypergeometric function, c1 = AiryAi(0) and c2 = -AiryAi'(0). - The two argument forms are used to represent the derivatives, so AiryAi(1,x) = D(AiryAi)(x) and AiryBi(1,x) = D(AiryBi)(x). Note that all higher derivatives can be written in terms of the 0'th and 1st derivatives. Note also that AiryAi(3,x^2) is the 3rd derivative of AiryAi(x) evaluated at x^2, and not the 3rd derivative of AiryAi(x^2). - The Airy functions are related to Bessel functions of order n/3 for n=-2,-1,1,2 (see the examples below). > convert(AiryAi(x), Bessel); 1/2 1/2 3/2 3 x BesselK(1/3, 2/3 x ) 1/3 -------------------------------- Pi > convert(AiryBi(1,x), Bessel); 1/2 3/2 3/2 1/3 x 3 (BesselI(-2/3, 2/3 x ) + BesselI(2/3, 2/3 x ))