# §9.4 Maclaurin Series

For $z\in\mathbb{C}$

 9.4.1 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ $\displaystyle=\mathop{\mathrm{Ai}\/}\nolimits\!\left(0\right)\left(1+\frac{1}{% 3!}z^{3}+\frac{1\cdot 4}{6!}z^{6}+\frac{1\cdot 4\cdot 7}{9!}z^{9}+\cdots\right% )+\mathop{\mathrm{Ai}\/}\nolimits'\!\left(0\right)\left(z+\frac{2}{4!}z^{4}+% \frac{2\cdot 5}{7!}z^{7}+\frac{2\cdot 5\cdot 8}{10!}z^{10}+\cdots\right),$ Symbols: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(\NVar{z}\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable A&S Ref: 10.4.2 (with 10.4.4 and 10.4.5) Permalink: http://dlmf.nist.gov/9.4.E1 Encodings: TeX, pMML, png See also: Annotations for 9.4 9.4.2 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits'\!\left(z\right)$ $\displaystyle=\mathop{\mathrm{Ai}\/}\nolimits'\!\left(0\right)\left(1+\frac{2}% {3!}z^{3}+\frac{2\cdot 5}{6!}z^{6}+\frac{2\cdot 5\cdot 8}{9!}z^{9}+\cdots% \right)+\mathop{\mathrm{Ai}\/}\nolimits\!\left(0\right)\left(\frac{1}{2!}z^{2}% +\frac{1\cdot 4}{5!}z^{5}+\frac{1\cdot 4\cdot 7}{8!}z^{8}+\cdots\right),$ Symbols: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(\NVar{z}\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable Permalink: http://dlmf.nist.gov/9.4.E2 Encodings: TeX, pMML, png See also: Annotations for 9.4 9.4.3 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ $\displaystyle=\mathop{\mathrm{Bi}\/}\nolimits\!\left(0\right)\left(1+\frac{1}{% 3!}z^{3}+\frac{1\cdot 4}{6!}z^{6}+\frac{1\cdot 4\cdot 7}{9!}z^{9}+\cdots\right% )+\mathop{\mathrm{Bi}\/}\nolimits'\!\left(0\right)\left(z+\frac{2}{4!}z^{4}+% \frac{2\cdot 5}{7!}z^{7}+\frac{2\cdot 5\cdot 8}{10!}z^{10}+\cdots\right),$ Symbols: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(\NVar{z}\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable A&S Ref: 10.4.3 (with 10.4.4 and 10.4.5) Referenced by: §9.12(vi) Permalink: http://dlmf.nist.gov/9.4.E3 Encodings: TeX, pMML, png See also: Annotations for 9.4 9.4.4 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits'\!\left(z\right)$ $\displaystyle=\mathop{\mathrm{Bi}\/}\nolimits'\!\left(0\right)\left(1+\frac{2}% {3!}z^{3}+\frac{2\cdot 5}{6!}z^{6}+\frac{2\cdot 5\cdot 8}{9!}z^{9}+\cdots% \right)+\mathop{\mathrm{Bi}\/}\nolimits\!\left(0\right)\left(\frac{1}{2!}z^{2}% +\frac{1\cdot 4}{5!}z^{5}+\frac{1\cdot 4\cdot 7}{8!}z^{8}+\cdots\right).$ Symbols: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(\NVar{z}\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable Permalink: http://dlmf.nist.gov/9.4.E4 Encodings: TeX, pMML, png See also: Annotations for 9.4