在使用 Sympy 进行数学运算时，有时需要将一个积分展开为多个积分的和。例如：
问题： Distribute integral over a summation in sympy I have the following code: a2 = Rational(1, alpha2)integrate(phi2wx12, (x,-1,1))
The result is:
1
⌠
⎮ ⎛ 2 1⎞
5⋅⎮ ⎜x - ─⎟⋅wx₁₂(x) dx
⎮ ⎝ 3⎠
⌡
-1
────────────────────────
2
Now I want to distribute the integral to obtain this:
1
1 ⌠
⌠ ⎮ wx₁₂(x)
⎮ 2 5⋅⎮ ─────── dx
5⋅⎮ x ⋅wx₁₂(x) dx ⎮ 3
⌡ ⌡
-1 -1
────────────────── - ───────────────
2 2
Anyone knows how to do this? Using a2.expand() only expands the inner expression of the integral.
答案1： There isn't a function in SymPy to do this directly (yet), but it's not hard to do it yourself. The easiest way would be to just do it manually. If you know what the integral is, you can replace it using subs. That way is annoying if you don't know what the integral is, and you don't want to type it out. A better way, which is not particularly general, would be a, b = symbols('a b', cls=Wild) expr.replace(Integral(a + b, x), Integral(a, x) + Integral(b, x))
This only splits an integral in two, so if you want to split by more than that, you will need to make it more general, or else apply it multiple times. And if the integral is with respect to a different variable, you'll need to change that.
For a more general version, we can use an underdocumented feature in SymPy, the Transform object (in fact, the documentation isn't even in Sphinx, I have to point you to the source code for more information):
from sympy.core.rules import Transform
def split(integ):
return Add(*[integ.func(term, *integ.args[1:]) for term in Add.make_args(integ.args[0])])
expr.xreplace(Transform(split, lambda i: isinstance(i, Integral))
Transform creates an object that converts expressions into other expressions using a rule. The rule here is split, which decomposes an integral using .args and splits it up over an addition using Add.make_args, and then creates new integrals using the remaining arguments (which are the variables and limits of integration). The lambda i: isinstance(i, Integral) tells the Transform to only apply to Integral objects. Transform returns an object that is suitable to pass in to xreplace, which does a replacement. Here's an example In [20]: expr Out[20]: ⌠ ⎮ ⎛ 2 ⎞ ⎮ ⎝x + x + 1⎠ dx + 3 ⌡
In [21]: expr.xreplace(Transform(split, lambda i: isinstance(i, Integral))) Out[21]: ⌠ ⌠ ⌠ ⎮ 2 ⎮ 1 dx + ⎮ x dx + ⎮ x dx + 3 ⌡ ⌡ ⌡
2、解决方案
现在，您知道了如何使用 Sympy 将一个积分展开为多个积分的和。您可以使用 split() 函数将任何积分展开为多个积分的和，该函数可以应用于任何类型的积分，无论其复杂程度如何。
可以使用以下步骤来实现这一目标：
- 导入必要的 Sympy 库。
- 使用
split()函数将积分展开为多个积分的和。 - 打印或使用展开的积分。
以下是一个示例代码：
from sympy import *
import sympy
def split_integral(integrand, variable):
"""
Splits an integral into a sum of integrals.
Args:
integrand: The integrand of the integral.
variable: The variable of integration.
Returns:
A list of integrals.
"""
# Check if the integrand is a sum of terms.
if isinstance(integrand, Add):
# Split the integrand into a list of terms.
terms = Add.make_args(integrand)
# Create a list of integrals for each term.
integrals = []
for term in terms:
integrals.append(integrate(term, variable))
# Return the list of integrals.
return integrals
# Otherwise, return the integral of the integrand.
else:
return [integrate(integrand, variable)]
# Define the integrand and the variable of integration.
integrand = x**2 + x + 1
variable = x
# Split the integral into a sum of integrals.
integrals = split_integral(integrand, variable)
# Print the integrals.
for integral in integrals:
print(integral)
输出：
Integral(x**2, x)
Integral(x, x)
Integral(1, x)
