§9.4 Maclaurin Series

Notes:: See Miller (1946, p. B17) and Olver (1997b, p. 54).
Keywords:: Airy functions
Referenced by:: §9.17(i)
Permalink:: http://dlmf.nist.gov/9.4

For $z\in\Complex$

9.4.1 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)=\mathop{\mathrm{Ai}\/}\nolimits\!\left(0\right)\left(1+\frac{1}{3!}z^{3}+\frac{1\cdot 4}{6!}z^{6}+\frac{1\cdot 4\cdot 7}{9!}z^{9}+\cdots\right)+{\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(0\right)\left(z+\frac{2}{4!}z^{4}+\frac{2\cdot 5}{7!}z^{7}+\frac{2\cdot 5\cdot 8}{10!}z^{{10}}+\cdots\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $!$ : $n!$ : factorial 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

9.4.2 ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(z\right)={\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(0\right)\left(1+\frac{2}{3!}z^{3}+\frac{2\cdot 5}{6!}z^{6}+\frac{2\cdot 5\cdot 8}{9!}z^{9}+\cdots\right)+\mathop{\mathrm{Ai}\/}\nolimits\!\left(0\right)\left(\frac{1}{2!}z^{2}+\frac{1\cdot 4}{5!}z^{5}+\frac{1\cdot 4\cdot 7}{8!}z^{8}+\cdots\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $!$ : $n!$ : factorial and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/9.4.E2
Encodings:: TeX, pMML, png

9.4.3 $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)=\mathop{\mathrm{Bi}\/}\nolimits\!\left(0\right)\left(1+\frac{1}{3!}z^{3}+\frac{1\cdot 4}{6!}z^{6}+\frac{1\cdot 4\cdot 7}{9!}z^{9}+\cdots\right)+{\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(0\right)\left(z+\frac{2}{4!}z^{4}+\frac{2\cdot 5}{7!}z^{7}+\frac{2\cdot 5\cdot 8}{10!}z^{{10}}+\cdots\right),$

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $!$ : $n!$ : factorial 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

9.4.4 ${\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(z\right)={\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(0\right)\left(1+\frac{2}{3!}z^{3}+\frac{2\cdot 5}{6!}z^{6}+\frac{2\cdot 5\cdot 8}{9!}z^{9}+\cdots\right)+\mathop{\mathrm{Bi}\/}\nolimits\!\left(0\right)\left(\frac{1}{2!}z^{2}+\frac{1\cdot 4}{5!}z^{5}+\frac{1\cdot 4\cdot 7}{8!}z^{8}+\cdots\right).$

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $!$ : $n!$ : factorial and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/9.4.E4
Encodings:: TeX, pMML, png