I am trying to numerically integrate the integral representation of $\operatorname{Ai}^2(x)$. The representation is
$$\operatorname{Ai}^2(t)=\frac{1}{4\pi^{3/2}i}\int_Ce^{-\frac{z^3}{12}-tz}z^{-1/2}dz.$$
Where the contour $C$ has upper portion $y_1(x)=\sqrt{3}\sqrt{x^2-4}$ and lower portion $y_2(x)=-\sqrt{3}\sqrt{x^2-4}$ where $x\in[2,\infty)$.
Since most of the integral is concentrated near the origin as, I can take the domain $x\in[2,10]$. Then I chop up the domain into $n$ bins.
$$\operatorname{Ai}^2(t)=\frac{1}{4\pi^{3/2}i}\int_Ce^{-\frac{z^3}{12}-tz}z^{-1/2}dz\approx\sum_{i=0}^n f((2+i\Delta x,y_i(2+i\Delta x)), t)\Delta y$$
where $f(z,t)=\frac{1}{4\pi^{3/2}i}e^{-\frac{z^3}{12}-tz}z^{-1/2}$. However, when I do this, the answer does not agree with the correct solution. Where have I gone wrong?
import mathimport numpy as npfrom matplotlib import pyplot as pltfrom scipy import specialimport numpy as npdef y1(x): return np.sqrt(3)*np.sqrt(x**2-4)def y2(x): return -np.sqrt(3)*np.sqrt(x**2-4)def f(z,t): return 1/(4*math.pi**(3/2)*(1j))*np.exp((z**3)/12-t*z)*(z**(-1/2))def integral(t): bins= 1000 # Upper contour x=np.linspace(2,10,bins) output = 0 for i in range(1, len(x)): dy = (y1(x[i])-y1(x[i-1])) output += f(complex(x[i],y1(x[i])),t)*dy # Lower contour for i in range(1, len(x))[::-1]: dy = (y2(x[i])-y2(x[i-1])) output += f(complex(x[i],y2(x[i])),t)*dy return outputt=np.linspace(-4,4,100)# Numerical Answer plt.plot(t, [integral(index) for index in t])# Actual answerplt.plot(t, [special.airy(index)[0]**2 for index in t])plt.ylim(-1, 2)Here is a publicly editable co-lab notebook with the running code
https://colab.research.google.com/drive/1PuibieIHpKKSbscHwZB0Gv9It9HcpfpF?usp=sharing
