有什么方法可以在Python中的单位球体表面整合球形谐波？

发布于 2025-01-26 01:25:51 字数 933 浏览 2 评论 0原文

我正在尝试将其集成到python中，

f1 = lambda phi, theta: ( np.conj(sph_harm(m1, l1, theta, phi)) \
                           * sph_harm(m2, l2, theta, phi)     #theta--> 0-2pi, phi-->0-pi
                  * np.sin(phi) / np.power(1+epsilon*np.cos(n*phi), l2))

coeff1 = (l1*l2*(l2+1)) /(2*l1*(l1+1))
result = coeff1 * term1
print(result)

其中 m1 ， l1 ， m2 ， l2 具有某些值。例如 m1，m2 = -2， l1，l2 = 2。
我还使用Mathematica检查了我从Python获得的值。它们不匹配不同的值。例如，对于上述 m1，l1，m2，l2 的值，从python i获得了 0.01673 ，而Mathematica给我 1.001433 。对于 l 和 m 的其他较高值，有时差异是巨大的。

然后，我尝试了高斯正交方法来数值计算我的结果。尽管它与Mathematica的结果相匹配的某些值 l 和 m ，但它并不适用于所有值。

很奇怪的是，我发现只有当我考虑np.sin（phi） / np.power（1+epsilon*np.cos（n*phi），l2））< / code> < / code> 。如果我只尝试整合球形谐波，则不会显示任何差异。您是否知道为什么会发生这种情况以及如何解决？

原文

I am trying to integrate this in Python

f1 = lambda phi, theta: ( np.conj(sph_harm(m1, l1, theta, phi)) \
                           * sph_harm(m2, l2, theta, phi)     #theta--> 0-2pi, phi-->0-pi
                  * np.sin(phi) / np.power(1+epsilon*np.cos(n*phi), l2))

coeff1 = (l1*l2*(l2+1)) /(2*l1*(l1+1))
result = coeff1 * term1
print(result)

where m1, l1, m2, l2 have certain values. For example m1,m2=-2, l1,l2=2.
I have also used Mathematica to check the values I get from python. They don't match for different values. For example, for the above mentioned values of m1,l1,m2,l2, from python I obtained 0.01673 while Mathematica gives me 1.001433. For other higher values of l and ms the difference is monumental sometimes.

Then, I have tried Gaussian quadrature method to numerically calculate my result. Though it matches for some values of l and m with the result from Mathematica, It does not hold for all values.

The weird thing is I found the problem occurring only when I consider np.sin(phi) / np.power(1+epsilon*np.cos(n*phi), l2)) in the integration. If I only try to integrate the spherical harmonics, it does not show any difference. Do you have any idea why this is happening and how to fix it?

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子栖 2025-02-02 01:25:51

请注意，在许多关于潜在理论的情况下，我更好地说明了类似的东西，

import numpy as np
from scipy.integrate import simps

xl = np.linspace( a, b, A ) 
yl = np.linspace( c, d, C )
xx, yy = np.meshgrid( xl, yl )
zz = xx.copy()
for i in range( len( yy ) ):
    for j in range( len( xx[0] ) ):
        x0 = xx[ i, j ]
        y0 = yy[ i, j ]
        zz[ i, j ] = f( x0, y0 )
result = simps( simps( zz, xl ), yl )

而不是使用dblquad。

Note in many cases on potential theory, I was better of with something like

import numpy as np
from scipy.integrate import simps

xl = np.linspace( a, b, A ) 
yl = np.linspace( c, d, C )
xx, yy = np.meshgrid( xl, yl )
zz = xx.copy()
for i in range( len( yy ) ):
    for j in range( len( xx[0] ) ):
        x0 = xx[ i, j ]
        y0 = yy[ i, j ]
        zz[ i, j ] = f( x0, y0 )
result = simps( simps( zz, xl ), yl )

rather than using dblquad.

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