为何Scipy贝塞尔函数求根不返回x=0处的根？

Let's break this down clearly:

First off, scipy.special.jn_zeros is explicitly designed to return only positive real zeros of the nth-order Bessel function of the first kind. That's the core reason you're not seeing x=0 in your output.

What's the full zero set for J₁(x)? #

The first-order Bessel function J₁(x) has zeros at:

• x=0 (since for all n ≥ 1, Jₙ(0) = 0)
• ±3.83170597, ±7.01558667, ±10.17346814, and so on

But jn_zeros(n, nt) returns the first nt positive zeros by definition. So when you call jn_zeros(1,2), you get the first two positive zeros: [3.83170597, 7.01558667]—which matches the actual output you're seeing.

Why do zeros appear symmetric along the x-axis? #

This ties to the symmetry property of Bessel functions:

• For odd n (like n=1 here): Jₙ(-x) = (-1)ⁿJₙ(x) = -Jₙ(x). So if x is a zero, -x will also be a zero (since J₁(-x) = -J₁(x) = 0).
• For even n: Jₙ(-x) = Jₙ(x), so zeros are symmetric around x=0 too.

The function just doesn't include negative zeros or x=0 by default.

If you need to include x=0 in your results #

You can manually prepend it to the array returned by jn_zeros:

import scipy.special as sp
import numpy as np

# Get first 2 positive zeros of J₁
positive_zeros = sp.jn_zeros(1, 2)
# Add x=0 to get the set you expected
full_zeros = np.concatenate(([0.], positive_zeros))
# Output: array([0., 3.83170597, 7.01558667])

内容的提问来源于stack exchange，提问作者WolcottR