A previous post of mine looked at the angles that graphs make when they cross. For example, sin() and cos(x) always cross with the same angle. The same holds for sin() and cos(kx) since the k simply rescales the x-axis.

The post ended with wondering about functions analogous to sine and cosine, such as Bessel functions. This post will look at that question in more detail. Specifically we'll look at the functions J_ν and Y_ν.

Because these two Bessel functions satisfy the same second order linear homogeneous differential equation, the Strum separation theorem says that their zeros are interlaced: between each pair of consecutive zeros of J_ν is exactly one zero of Y_ν, and between each pair of consecutive zeros of Y_ν there is exactly one zero of

In the following Python code, we find zeros of J_ν, then look in between for places where J_ν and Y_ν cross. Next we find the angle the two curves make at each intersection and plot the angles.

    from scipy.special import jn_zeros, jv, yv
    from scipy.optimize import bisect
    from numpy import empty, linspace, arccos
    import matplotlib.pyplot as plt

    n = 3 # bessel function order
    N = 100 # number of zeros

    z = jn_zeros(n, N) # Zeros of J_n
    crossings = empty(N-1)

    f = lambda x: jv(n,x) - yv(n,x)    
    for i in range(N-1):
        crossings[i] = bisect(f, z[i], z[i+1])

    def angle(n, x):
        # Derivatives of J_nu and Y_nu
        dj = 0.5*(jv(n-1,x) - jv(n+1,x))
        dy = 0.5*(yv(n-1,x) - yv(n+1,x))

        top = 1 + dj*dy
        bottom = ((1 + dj**2)*(1 + dy**2))**0.5
        return arccos(top/bottom)

    y = angle(n, crossings)
    plt.plot(y)
    plt.xlabel("Crossing number")
    plt.ylabel("Angle in radians")
    plt.show()

This shows that the angles steadily decrease, apparently quadratically.

This quadratic behavior is what we should expect from the asymptotics of J_ν and Y_ν: For large arguments they act like shifted and rescaled versions of sin(x)/√x. So if we looked at √xJ_ν and √xY_ν rather than J_ν and Y_ν we'd expect the angles to reach some positive asymptote, and they do, as shown below.