特殊函数的渐近展开式

Template:Unreferenced 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

$BesselY(3,x),\approx sqrt(2)*cos(x-(1/4)*Pi)*sqrt(1/x)/sqrt(Pi)+(35/8)*sqrt(2)*cos(x+(1/4)*Pi)*(1/x{)}^{(}3/2)/sqrt(Pi)-(945/128)*sqrt(2)*cos(x-(1/4)*Pi)*(1/x{)}^{(}5/2)/sqrt(Pi)-(3465/1024)*sqrt(2)*cos(x+(1/4)*Pi)*(1/x{)}^{(}7/2)/sqrt(Pi)-(45045/32768)*sqrt(2)*cos(x-(1/4)*Pi)*(1/x{)}^{(}9/2)/sqrt(Pi)+O((1/x{)}^{(}11/2))$ $\mathrm{\Gamma }(z)\approx (ln(z)-1)*z+ln(\sqrt{(}2)*\sqrt{(}\pi ))-(1/2)*ln(z)+1/(12*z)-1/(360*z^3)+1/(1260*z^5)-1/(1680*z^7)+O(1/z^9)$ 误差函数

• $erf(x)\approx 1+(-1/(\sqrt{(}\pi )*x)+1/(2*\sqrt{(}\pi )*x^3)-3/(4*\sqrt{(}\pi )*x^5)+15/(8*\sqrt{(}\pi )*x^7)-105/(16*\sqrt{(}\pi )*x^9)+O(1/x^{1}1))/exp(x^2)$

斐涅尔函数 $FresnelC(x)\approx 1/2+sin((1/2)*\pi *x^2)/(\pi *x)-cos((1/2)*\pi *x^2)/({\pi }^2*x^3)-3*sin((1/2)*\pi *x^2)/({\pi }^3*x^5)+15*cos((1/2)*\pi *x^2)/({\pi }^4*x^7)+105*sin((1/2)*\pi *x^2)/({\pi }^5*x^9)$

• $FresnelS(x)\approx 1/2-cos((1/2)*\pi *x^2)/(\pi *x)-sin((1/2)*\pi *x^2)/({\pi }^2*x^3)+3*cos((1/2)*\pi *x^2)/({\pi }^3*x^5)+15*sin((1/2)*\pi *x^2)/({\pi }^4*x^7)-105*cos((1/2)*\pi *x^2)/({\pi }^5*x^9)+O(1/x^{1}0)$