# §9.4 Maclaurin Series

For $z\in\mathbb{C}$

 9.4.1 $\displaystyle\operatorname{Ai}\left(z\right)$ $\displaystyle=\operatorname{Ai}\left(0\right)\left(1+\frac{1}{3!}z^{3}+\frac{1% \cdot 4}{6!}z^{6}+\frac{1\cdot 4\cdot 7}{9!}z^{9}+\cdots\right)+\operatorname{% Ai}'\left(0\right)\left(z+\frac{2}{4!}z^{4}+\frac{2\cdot 5}{7!}z^{7}+\frac{2% \cdot 5\cdot 8}{10!}z^{10}+\cdots\right),$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable Source: Olver (1997b, p. 54) 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 and Ch.9 9.4.2 $\displaystyle\operatorname{Ai}'\left(z\right)$ $\displaystyle=\operatorname{Ai}'\left(0\right)\left(1+\frac{2}{3!}z^{3}+\frac{% 2\cdot 5}{6!}z^{6}+\frac{2\cdot 5\cdot 8}{9!}z^{9}+\cdots\right)+\operatorname% {Ai}\left(0\right)\left(\frac{1}{2!}z^{2}+\frac{1\cdot 4}{5!}z^{5}+\frac{1% \cdot 4\cdot 7}{8!}z^{8}+\cdots\right),$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable Proof sketch: Use methods of Olver (1997b, p. 54). Permalink: http://dlmf.nist.gov/9.4.E2 Encodings: TeX, pMML, png See also: Annotations for §9.4 and Ch.9 9.4.3 $\displaystyle\operatorname{Bi}\left(z\right)$ $\displaystyle=\operatorname{Bi}\left(0\right)\left(1+\frac{1}{3!}z^{3}+\frac{1% \cdot 4}{6!}z^{6}+\frac{1\cdot 4\cdot 7}{9!}z^{9}+\cdots\right)+\operatorname{% Bi}'\left(0\right)\left(z+\frac{2}{4!}z^{4}+\frac{2\cdot 5}{7!}z^{7}+\frac{2% \cdot 5\cdot 8}{10!}z^{10}+\cdots\right),$ ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable Proof sketch: Use methods of Olver (1997b, p. 54). A&S Ref: 10.4.3 (with 10.4.4 and 10.4.5) Referenced by: (9.12.15) Permalink: http://dlmf.nist.gov/9.4.E3 Encodings: TeX, pMML, png See also: Annotations for §9.4 and Ch.9 9.4.4 $\displaystyle\operatorname{Bi}'\left(z\right)$ $\displaystyle=\operatorname{Bi}'\left(0\right)\left(1+\frac{2}{3!}z^{3}+\frac{% 2\cdot 5}{6!}z^{6}+\frac{2\cdot 5\cdot 8}{9!}z^{9}+\cdots\right)+\operatorname% {Bi}\left(0\right)\left(\frac{1}{2!}z^{2}+\frac{1\cdot 4}{5!}z^{5}+\frac{1% \cdot 4\cdot 7}{8!}z^{8}+\cdots\right).$ ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable Proof sketch: Use methods of Olver (1997b, p. 54). Permalink: http://dlmf.nist.gov/9.4.E4 Encodings: TeX, pMML, png See also: Annotations for §9.4 and Ch.9