ESF - The Encyclopedia of Special Functions

About Equation WM.2.1.1.1

$\begin{equation*} \begin{split} & \operatorname{WM} _{\mu , \nu} (x)\approx x^{\Bigl(\nu + \frac{1}{2}\Bigr)} \Biggl(1 - \frac{\mu x}{2 \nu + 1} + \frac{\bigl(2 \nu + 1 + 4 \mu^{2}\bigr) x^{2}}{(8 \nu + 8) (4 \nu + 2)} + \\ & \quad{}\quad{}\frac{-\mu \bigl(6 \nu + 5 + 4 \mu^{2}\bigr) x^{3}}{12 (8 \nu + 12) (2 \nu + 1) (\nu + 1)} + \\ & \quad{}\quad{}\frac{\bigl(12 \nu^{2} + 24 \nu + 9 + 48 \mu^{2} \nu + 56 \mu^{2} + 16 \mu^{4}\bigr) x^{4}}{192 (8 \nu + 16) (2 \nu + 1) (\nu + 1) (2 \nu + 3)} + \\ & \quad{}\quad{}\frac{-\mu \bigl(60 \nu^{2} + 160 \nu + 89 + 80 \mu^{2} \nu + 120 \mu^{2} + 16 \mu^{4}\bigr) x^{5}}{1920 (8 \nu + 20) (2 \nu + 1) (\nu + 1) (2 \nu + 3) (\nu + 2)} + \Bigl(\bigl( \\ & \quad{}\quad{}120 \nu^{3} + 540 \nu^{2} + 690 \nu + 225 + 720 \mu^{2} \nu^{2} + 2400 \mu^{2} \nu + 1756 \mu^{2} \\ & \quad{}\quad{}+ 480 \mu^{4} \nu + 880 \mu^{4} + 64 \mu^{6}\bigr) x^{6}\Bigr)\bigg/ \\ & \quad{}\quad{}\bigl(46080 (8 \nu + 24) (2 \nu + 1) (\nu + 1) (2 \nu + 3) (\nu + 2) (2 \nu + 5)\bigr) + \Bigl(-\mu \bigl( \\ & \quad{}\quad{}840 \nu^{3} + 4620 \nu^{2} + 7518 \nu + 3429 + 1680 \mu^{2} \nu^{2} + 6720 \mu^{2} \nu + \\ & \quad{}\quad{}6076 \mu^{2} + 672 \mu^{4} \nu + 1456 \mu^{4} + 64 \mu^{6}\bigr) x^{7}\Bigr)\bigg/\bigl(645120 (8 \nu + 28) \\ & \quad{}\quad{}(2 \nu + 1) (\nu + 1) (2 \nu + 3) (\nu + 2) (2 \nu + 5) (\nu + 3)\bigr)\ldots\Biggr) \end{split} \end{equation*}$

Absolute reference: WM:asympt:0:terms

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\operatorname{WM} _{\mu , \nu} (x)\approx x^{\Bigl(\nu + \frac{1}{2}\Bigr)} \Biggl(1 - \frac{\mu x}{2 \nu + 1} + \frac{\bigl(2 \nu + 1 + 4 \mu^{2}\bigr) x^{2}}{(8 \nu + 8) (4 \nu + 2)} + \\ \quad{}\quad{}\frac{-\mu \bigl(6 \nu + 5 + 4 \mu^{2}\bigr) x^{3}}{12 (8 \nu + 12) (2 \nu + 1) (\nu + 1)} + \\ \quad{}\quad{}\frac{\bigl(12 \nu^{2} + 24 \nu + 9 + 48 \mu^{2} \nu + 56 \mu^{2} + 16 \mu^{4}\bigr) x^{4}}{192 (8 \nu + 16) (2 \nu + 1) (\nu + 1) (2 \nu + 3)} + \\ \quad{}\quad{}\frac{-\mu \bigl(60 \nu^{2} + 160 \nu + 89 + 80 \mu^{2} \nu + 120 \mu^{2} + 16 \mu^{4}\bigr) x^{5}}{1920 (8 \nu + 20) (2 \nu + 1) (\nu + 1) (2 \nu + 3) (\nu + 2)} + \Bigl(\bigl( \\ \quad{}\quad{}120 \nu^{3} + 540 \nu^{2} + 690 \nu + 225 + 720 \mu^{2} \nu^{2} + 2400 \mu^{2} \nu + 1756 \mu^{2} \\ \quad{}\quad{}+ 480 \mu^{4} \nu + 880 \mu^{4} + 64 \mu^{6}\bigr) x^{6}\Bigr)\bigg/ \\ \quad{}\quad{}\bigl(46080 (8 \nu + 24) (2 \nu + 1) (\nu + 1) (2 \nu + 3) (\nu + 2) (2 \nu + 5)\bigr) + \Bigl(-\mu \bigl( \\ \quad{}\quad{}840 \nu^{3} + 4620 \nu^{2} + 7518 \nu + 3429 + 1680 \mu^{2} \nu^{2} + 6720 \mu^{2} \nu + \\ \quad{}\quad{}6076 \mu^{2} + 672 \mu^{4} \nu + 1456 \mu^{4} + 64 \mu^{6}\bigr) x^{7}\Bigr)\bigg/\bigl(645120 (8 \nu + 28) \\ \quad{}\quad{}(2 \nu + 1) (\nu + 1) (2 \nu + 3) (\nu + 2) (2 \nu + 5) (\nu + 3)\bigr)\ldots\Biggr)

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`&sepsequence`(`&nothing`,`&mathseq`(`&operatorname`(WM),`&subscript`(`&mathseq`(mu,comma,nu)))(x),approx,nullspace,x^(nu+1/2)*`&delimiter`(leftparens,rightparens,false,`&sepsequence`(`&nothing`,1-mu/(2*nu+1)*x+1/2*(1+4*mu^2/(2*nu+1))/(8*nu+8)*x^2+1/3*(-mu/(2*nu+1)-2*mu*(1+4*mu^2/(2*nu+1))/(8*nu+8))/(8*nu+12)*x^3+1/4*(1/2*(1+4*mu^2/(2*nu+1))/(8*nu+8)-4/3*mu*(-mu/(2*nu+1)-2*mu*(1+4*mu^2/(2*nu+1))/(8*nu+8))/(8*nu+12))/(8*nu+16)*x^4+1/5*(1/3*(-mu/(2*nu+1)-2*mu*(1+4*mu^2/(2*nu+1))/(8*nu+8))/(8*nu+12)-mu*(1/2*(1+4*mu^2/(2*nu+1))/(8*nu+8)-4/3*mu*(-mu/(2*nu+1)-2*mu*(1+4*mu^2/(2*nu+1))/(8*nu+8))/(8*nu+12))/(8*nu+16))/(8*nu+20)*x^5+1/6*(1/4*(1/2*(1+4*mu^2/(2*nu+1))/(8*nu+8)-4/3*mu*(-mu/(2*nu+1)-2*mu*(1+4*mu^2/(2*nu+1))/(8*nu+8))/(8*nu+12))/(8*nu+16)-4/5*mu*(1/3*(-mu/(2*nu+1)-2*mu*(1+4*mu^2/(2*nu+1))/(8*nu+8))/(8*nu+12)-mu*(1/2*(1+4*mu^2/(2*nu+1))/(8*nu+8)-4/3*mu*(-mu/(2*nu+1)-2*mu*(1+4*mu^2/(2*nu+1))/(8*nu+8))/(8*nu+12))/(8*nu+16))/(8*nu+20))/(8*nu+24)*x^6+1/7*(1/5*(1/3*(-mu/(2*nu+1)-2*mu*(1+4*mu^2/(2*nu+1))/(8*nu+8))/(8*nu+12)-mu*(1/2*(1+4*mu^2/(2*nu+1))/(8*nu+8)-4/3*mu*(-mu/(2*nu+1)-2*mu*(1+4*mu^2/(2*nu+1))/(8*nu+8))/(8*nu+12))/(8*nu+16))/(8*nu+20)-2/3*mu*(1/4*(1/2*(1+4*mu^2/(2*nu+1))/(8*nu+8)-4/3*mu*(-mu/(2*nu+1)-2*mu*(1+4*mu^2/(2*nu+1))/(8*nu+8))/(8*nu+12))/(8*nu+16)-4/5*mu*(1/3*(-mu/(2*nu+1)-2*mu*(1+4*mu^2/(2*nu+1))/(8*nu+8))/(8*nu+12)-mu*(1/2*(1+4*mu^2/(2*nu+1))/(8*nu+8)-4/3*mu*(-mu/(2*nu+1)-2*mu*(1+4*mu^2/(2*nu+1))/(8*nu+8))/(8*nu+12))/(8*nu+16))/(8*nu+20))/(8*nu+24))/(8*nu+28)*x^7,ldots)))