特殊函數的漸近展開式

特殊函數的漸近展開式

安格爾函數

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

艾瑞函數

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

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

貝索函數

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

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

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

Γ函數

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

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

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

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