See: here

For example Integral representations AiryAi[z] is :

AiryAi[z] == (1/Pi) Integrate[Cos[t^3/3 + z t], {t, 0, Infinity}] /; Im[z] == 0

Or use this commands:

MathematicalFunctionData["AiryAi", "SeriesRepresentations"]
MathematicalFunctionData["AiryAi", "IntegralRepresentations"]
MathematicalFunctionData["AiryAi", "LimitRepresentations"]
MathematicalFunctionData["AiryAi", "AlternativeRepresentations"]

Regards M.I.

Binomial[n, p]) to an algebraic format ?.Probably you want:

FullSimplify[Binomial[n, p] // FunctionExpand, Assumptions -> {n, p} \[Element] PositiveIntegers]
(*n!/(p! Gamma[1 + n - p])*)
AiryAi[z] // FunctionExpand(*Dosen't work ! *)

For integral where: Im[z] == 0 that's mean,formula works only for: $z\in \mathbb{R}$

You can plot the integral form, see code below:

AIR[z_?NumericQ] := (1/Pi) *NIntegrate[Cos[t^3/3 + z t], {t, 0, Infinity}]
Plot[{AIR[z], AiryAi[z]}, {z, -2, 2}, PlotStyle -> {Red, {Black, Dashed}}, PlotLegends -> "Expressions"]

Regards M.I.