查看“︁特殊函数的渐近展开式”︁的源代码

{{unreferenced|time=2017-05-12T10:32:50+00:00}}
'''特殊函数的渐近展开式'''

;[[安格尔函数]]
<math>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))}</math>

;[[艾瑞函数]]
<math>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))</math>
*<math>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))</math>
;[[贝塞尔函数]]
<math>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))}</math>

<math>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))}</math>

<math>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))}</math>
;[[Γ函数]]

<math> 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))}</math>
<math>\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)</math>
[[误差函数]]
*<math>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)}</math>
[[斐涅尔函数]]
<math>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)</math>
*<math>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^10)}</math>

[[Category:特殊函数]]
[[Category:渐近分析]]