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

$AngerJ(3,x)\approx \sqrt{(}2)\ast cos(x+(1/4)\ast Pi)\ast \sqrt{(}1/x)/\sqrt{(}Pi)-(35/8)\ast \sqrt{(}2)\ast cos(x-(1/4)\ast Pi)\ast (1/x{)}^{(}3/2)/\sqrt{(}\pi )-(945/128)\ast \sqrt{(}2)\ast cos(x+(1/4)\ast \pi )\ast (1/x{)}^{(}5/2)/\sqrt{(}\pi )+O((1/x{)}^{(}7/2))$

艾瑞函数

$AiryAi(z)\approx (1/2)\ast exp(-(2/3)\ast z^{(}3/2))\ast (1/z{)}^{(}1/4)/\sqrt{(}\pi )-(5/96)\ast exp(-(2/3)\ast z^{(}3/2))\ast (1/z{)}^{(}7/4)/\sqrt{(}\pi )+(385/9216)\ast exp(-(2/3)\ast z^{(}3/2))\ast (1/z{)}^{(}13/4)/\sqrt{(}\pi )+O((1/z{)}^{(}19/4))$

• $AiryBi(z)\approx exp((2/3)\ast z^{(}3/2))\ast (1/z{)}^{(}1/4)/\sqrt{(}\pi )+(5/48)\ast exp((2/3)\ast z^{(}3/2))\ast (1/z{)}^{(}7/4)/\sqrt{(}\pi )+(385/4608)\ast exp((2/3)\ast z^{(}3/2))\ast (1/z{)}^{(}13/4)/\sqrt{(}\pi )+O((1/z{)}^{(}19/4))$

贝塞尔函数

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

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

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

Γ函数

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

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

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

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