Formula:5866

Digital Library of Mathematical Functions ID 15.12.E9

${\displaystyle{\displaystyle(z+1)^{3\lambda/2}(2\lambda)^{c-1}\mathbf{F}\left({a+\lambda,b+2\lambda\atop c};-z\right)={\lambda^{-1/3}\left(e^{\pi\mathrm{i}(a-c+\lambda+(1/3))}\mathrm{Ai}\left(e^{-\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)+e^{\pi\mathrm{i}(c-a-\lambda-(1/3))}\mathrm{Ai}\left(e^{\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)\right)\left(a_{0}(\zeta)+O(\lambda^{-1})\right)}+\lambda^{-2/3}\left(e^{\pi\mathrm{i}(a-c+\lambda+(2/3))}\mathrm{Ai}'\left(e^{-\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)+e^{\pi\mathrm{i}(c-a-\lambda-(2/3))}\mathrm{Ai}'\left(e^{\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)\right)\left(a_{1}(\zeta)+O(\lambda^{-1})\right),}}$

Symbols List

• ${\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}$:
• ${\displaystyle{\displaystyle O\left(\NVar{x}\right)}}$: order not exceeding
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}$: Olver’s hypergeometric function
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle a}}$: real or complex parameter
• ${\displaystyle{\displaystyle b}}$: real or complex parameter
• ${\displaystyle{\displaystyle c}}$: real or complex parameter
• ${\displaystyle{\displaystyle\zeta}}$: change of variable
• ${\displaystyle{\displaystyle\beta}}$: β
• ${\displaystyle{\displaystyle a_{j}(\zeta)}}$: function
 

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