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 A n g e r J ( 3 , x ) ≈ ( 2 ) ∗ c o s ( x + ( 1 / 4 ) ∗ P i ) ∗ ( 1 / x ) / ( P i ) − ( 35 / 8 ) ∗ ( 2 ) ∗ c o s ( x − ( 1 / 4 ) ∗ P i ) ∗ ( 1 / x ) ( 3 / 2 ) / ( π ) − ( 945 / 128 ) ∗ ( 2 ) ∗ c o s ( x + ( 1 / 4 ) ∗ π ) ∗ ( 1 / x ) ( 5 / 2 ) / ( π ) + O ( ( 1 / x ) ( 7 / 2 ) ) {\displaystyle 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 A i r y A i ( z ) ≈ ( 1 / 2 ) ∗ e x p ( − ( 2 / 3 ) ∗ z ( 3 / 2 ) ) ∗ ( 1 / z ) ( 1 / 4 ) / ( π ) − ( 5 / 96 ) ∗ e x p ( − ( 2 / 3 ) ∗ z ( 3 / 2 ) ) ∗ ( 1 / z ) ( 7 / 4 ) / ( π ) + ( 385 / 9216 ) ∗ e x p ( − ( 2 / 3 ) ∗ z ( 3 / 2 ) ) ∗ ( 1 / z ) ( 13 / 4 ) / ( π ) + O ( ( 1 / z ) ( 19 / 4 ) ) {\displaystyle 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))} A i r y B i ( z ) ≈ e x p ( ( 2 / 3 ) ∗ z ( 3 / 2 ) ) ∗ ( 1 / z ) ( 1 / 4 ) / ( π ) + ( 5 / 48 ) ∗ e x p ( ( 2 / 3 ) ∗ z ( 3 / 2 ) ) ∗ ( 1 / z ) ( 7 / 4 ) / ( π ) + ( 385 / 4608 ) ∗ e x p ( ( 2 / 3 ) ∗ z ( 3 / 2 ) ) ∗ ( 1 / z ) ( 13 / 4 ) / ( π ) + O ( ( 1 / z ) ( 19 / 4 ) ) {\displaystyle 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 B e s s e l I ( 3 , x ) ≈ ( 1 / 2 ) ∗ s q r t ( 2 ) ∗ e x p ( x ) ∗ s q r t ( 1 / x ) / s q r t ( P i ) − ( 35 / 16 ) ∗ s q r t ( 2 ) ∗ e x p ( x ) ∗ ( 1 / x ) ( 3 / 2 ) / s q r t ( P i ) + ( 945 / 256 ) ∗ s q r t ( 2 ) ∗ e x p ( x ) ∗ ( 1 / x ) ( 5 / 2 ) / s q r t ( P i ) − ( 3465 / 2048 ) ∗ s q r t ( 2 ) ∗ e x p ( x ) ∗ ( 1 / x ) ( 7 / 2 ) / s q r t ( P i ) − ( 45045 / 65536 ) ∗ s q r t ( 2 ) ∗ e x p ( x ) ∗ ( 1 / x ) ( 9 / 2 ) / s q r t ( P i ) + O ( ( 1 / x ) ( 11 / 2 ) ) {\displaystyle 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))}} B e s s e l J ( 3 , x ) ≈ ( 2 ) ∗ c o s ( x + ( 1 / 4 ) ∗ P i ) ∗ ( 1 / x ) / ( π ) − ( 35 / 8 ) ∗ ( 2 ) ∗ c o s ( x − ( 1 / 4 ) ∗ π ) ∗ ( 1 / x ) ( 3 / 2 ) / ( π ) − ( 945 / 128 ) ∗ ( 2 ) ∗ c o s ( x + ( 1 / 4 ) ∗ π ) ∗ ( 1 / x ) ( 5 / 2 ) / ( π ) + ( 3465 / 1024 ) ∗ s q r t ( 2 ) ∗ c o s ( x − ( 1 / 4 ) ∗ π ) ∗ ( 1 / x ) ( 7 / 2 ) / ( π ) − ( 45045 / 32768 ) ∗ ( 2 ) ∗ c o s ( x + ( 1 / 4 ) ∗ π ) ∗ ( 1 / x ) ( 9 / 2 ) / ( π ) + O ( ( 1 / x ) ( 11 / 2 ) ) {\displaystyle 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))}} B e s s e l K ( 3 , ) ≈ ( 1 / 2 ) ∗ s q r t ( 2 ) ∗ s q r t ( P i ) ∗ e x p ( − x ) ∗ s q r t ( 1 / x ) + ( 35 / 16 ) ∗ s q r t ( 2 ) ∗ s q r t ( P i ) ∗ e x p ( − x ) ∗ ( 1 / x ) ( 3 / 2 ) + ( 945 / 256 ) ∗ s q r t ( 2 ) ∗ s q r t ( P i ) ∗ e x p ( − x ) ∗ ( 1 / x ) ( 5 / 2 ) + ( 3465 / 2048 ) ∗ s q r t ( 2 ) ∗ s q r t ( P i ) ∗ e x p ( − x ) ∗ ( 1 / x ) ( 7 / 2 ) − ( 45045 / 65536 ) ∗ s q r t ( 2 ) ∗ s q r t ( P i ) ∗ e x p ( − x ) ∗ ( 1 / x ) ( 9 / 2 ) + O ( ( 1 / x ) ( 11 / 2 ) ) {\displaystyle 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 B e s s e l Y ( 3 , x ) , ≈ s q r t ( 2 ) ∗ c o s ( x − ( 1 / 4 ) ∗ P i ) ∗ s q r t ( 1 / x ) / s q r t ( P i ) + ( 35 / 8 ) ∗ s q r t ( 2 ) ∗ c o s ( x + ( 1 / 4 ) ∗ P i ) ∗ ( 1 / x ) ( 3 / 2 ) / s q r t ( P i ) − ( 945 / 128 ) ∗ s q r t ( 2 ) ∗ c o s ( x − ( 1 / 4 ) ∗ P i ) ∗ ( 1 / x ) ( 5 / 2 ) / s q r t ( P i ) − ( 3465 / 1024 ) ∗ s q r t ( 2 ) ∗ c o s ( x + ( 1 / 4 ) ∗ P i ) ∗ ( 1 / x ) ( 7 / 2 ) / s q r t ( P i ) − ( 45045 / 32768 ) ∗ s q r t ( 2 ) ∗ c o s ( x − ( 1 / 4 ) ∗ P i ) ∗ ( 1 / x ) ( 9 / 2 ) / s q r t ( P i ) + O ( ( 1 / x ) ( 11 / 2 ) ) {\displaystyle 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))}} Γ ( z ) ≈ ( l n ( z ) − 1 ) ∗ z + l n ( ( 2 ) ∗ ( π ) ) − ( 1 / 2 ) ∗ l n ( z ) + 1 / ( 12 ∗ z ) − 1 / ( 360 ∗ z 3 ) + 1 / ( 1260 ∗ z 5 ) − 1 / ( 1680 ∗ z 7 ) + O ( 1 / z 9 ) {\displaystyle \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 e r f ( x ) ≈ 1 + ( − 1 / ( ( π ) ∗ x ) + 1 / ( 2 ∗ ( π ) ∗ x 3 ) − 3 / ( 4 ∗ ( π ) ∗ x 5 ) + 15 / ( 8 ∗ ( π ) ∗ x 7 ) − 105 / ( 16 ∗ ( π ) ∗ x 9 ) + O ( 1 / x 1 1 ) ) / e x p ( x 2 ) {\displaystyle 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 F r e s n e l C ( x ) ≈ 1 / 2 + s i n ( ( 1 / 2 ) ∗ π ∗ x 2 ) / ( π ∗ x ) − c o s ( ( 1 / 2 ) ∗ π ∗ x 2 ) / ( π 2 ∗ x 3 ) − 3 ∗ s i n ( ( 1 / 2 ) ∗ π ∗ x 2 ) / ( π 3 ∗ x 5 ) + 15 ∗ c o s ( ( 1 / 2 ) ∗ π ∗ x 2 ) / ( π 4 ∗ x 7 ) + 105 ∗ s i n ( ( 1 / 2 ) ∗ π ∗ x 2 ) / ( π 5 ∗ x 9 ) {\displaystyle 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})} F r e s n e l S ( x ) ≈ 1 / 2 − c o s ( ( 1 / 2 ) ∗ π ∗ x 2 ) / ( π ∗ x ) − s i n ( ( 1 / 2 ) ∗ π ∗ x 2 ) / ( π 2 ∗ x 3 ) + 3 ∗ c o s ( ( 1 / 2 ) ∗ π ∗ x 2 ) / ( π 3 ∗ x 5 ) + 15 ∗ s i n ( ( 1 / 2 ) ∗ π ∗ x 2 ) / ( π 4 ∗ x 7 ) − 105 ∗ c o s ( ( 1 / 2 ) ∗ π ∗ x 2 ) / ( π 5 ∗ x 9 ) + O ( 1 / x 1 0 ) {\displaystyle 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)}}