特殊函数的渐近展开式

特殊函数的渐近展开式

;安格尔函数
AngerJ(3,x) \approx {\sqrt(2)cos(x+(1/4)Pi)\sqrt(1/x)/\sqrt(Pi)-(35/8)\sqrt(2)cos(x-(1/4)Pi)(1/x)^(3/2)/\sqrt(\pi)-(945/128)\sqrt(2)cos(x+(1/4)\pi)*(1/x)^(5/2)/\sqrt(\pi)+O((1/x)^(7/2))}

;艾瑞函数
AiryAi(z)\approx (1/2)exp(-(2/3)z^(3/2))(1/z)^(1/4)/\sqrt(\pi)-(5/96)exp(-(2/3)z^(3/2))(1/z)^(7/4)/\sqrt(\pi)+(385/9216)exp(-(2/3)z^(3/2))*(1/z)^(13/4)/\sqrt(\pi)+O((1/z)^(19/4))
AiryBi(z) \approx exp((2/3)z^(3/2))(1/z)^(1/4)/\sqrt(\pi)+(5/48)exp((2/3)z^(3/2))(1/z)^(7/4)/\sqrt(\pi)+(385/4608)exp((2/3)z^(3/2))*(1/z)^(13/4)/\sqrt(\pi)+O((1/z)^(19/4))
;贝塞尔函数
BesselI(3,x) \approx {(1/2)sqrt(2)exp(x)sqrt(1/x)/sqrt(Pi)-(35/16)sqrt(2)exp(x)(1/x)^(3/2)/sqrt(Pi)+(945/256)sqrt(2)exp(x)(1/x)^(5/2)/sqrt(Pi)-(3465/2048)sqrt(2)exp(x)(1/x)^(7/2)/sqrt(Pi)-(45045/65536)sqrt(2)exp(x)*(1/x)^(9/2)/sqrt(Pi)+O((1/x)^(11/2))}

BesselJ(3,x) \approx {\sqrt(2)cos(x+(1/4)Pi)\sqrt(1/x)/\sqrt(\pi)-(35/8)\sqrt(2)cos(x-(1/4)\pi)(1/x)^(3/2)/\sqrt(\pi)-(945/128)\sqrt(2)cos(x+(1/4)\pi)(1/x)^(5/2)/\sqrt(\pi)+(3465/1024)sqrt(2)cos(x-(1/4)\pi)(1/x)^(7/2)/\sqrt(\pi)-(45045/32768)\sqrt(2)cos(x+(1/4)\pi)*(1/x)^(9/2)/\sqrt(\pi)+O((1/x)^(11/2))}

BesselK(3,) \approx {(1/2)sqrt(2)sqrt(Pi)exp(-x)sqrt(1/x)+(35/16)sqrt(2)sqrt(Pi)exp(-x)(1/x)^(3/2)+(945/256)sqrt(2)sqrt(Pi)exp(-x)(1/x)^(5/2)+(3465/2048)sqrt(2)sqrt(Pi)exp(-x)(1/x)^(7/2)-(45045/65536)sqrt(2)sqrt(Pi)exp(-x)(1/x)^(9/2)+O((1/x)^(11/2))}
;Γ函数

BesselY(3,x), \approx {sqrt(2)cos(x-(1/4)Pi)sqrt(1/x)/sqrt(Pi)+(35/8)sqrt(2)cos(x+(1/4)Pi)(1/x)^(3/2)/sqrt(Pi)-(945/128)sqrt(2)cos(x-(1/4)Pi)(1/x)^(5/2)/sqrt(Pi)-(3465/1024)sqrt(2)cos(x+(1/4)Pi)(1/x)^(7/2)/sqrt(Pi)-(45045/32768)sqrt(2)cos(x-(1/4)Pi)*(1/x)^(9/2)/sqrt(Pi)+O((1/x)^(11/2))}
\Gamma(z) \approx (ln(z)-1)z+ln(\sqrt(2)\sqrt(\pi))-(1/2)ln(z)+1/(12z)-1/(360z^3)+1/(1260z^5)-1/(1680*z^7)+O(1/z^9)
误差函数
erf(x) \approx {1+(-1/(\sqrt(\pi)x)+1/(2\sqrt(\pi)x^3)-3/(4\sqrt(\pi)x^5)+15/(8\sqrt(\pi)x^7)-105/(16\sqrt(\pi)x^9)+O(1/x^11))/exp(x^2)}
斐涅尔函数
FresnelC(x) \approx 1/2+sin((1/2)\pix^2)/(\pix)-cos((1/2)\pix^2)/(\pi^2x^3)-3sin((1/2)\pix^2)/(\pi^3x^5)+15cos((1/2)\pix^2)/(\pi^4x^7)+105sin((1/2)\pix^2)/(\pi^5x^9)
FresnelS(x) \approx {1/2-cos((1/2)\pix^2)/(\pix)-sin((1/2)\pix^2)/(\pi^2x^3)+3cos((1/2)\pix^2)/(\pi^3x^5)+15sin((1/2)\pix^2)/(\pi^4x^7)-105cos((1/2)\pix^2)/(\pi^5*x^9)+O(1/x^10)}