# 方法一：pip 安装（推荐）
pip install sympy

# 方法二：conda 安装
conda install -c anaconda sympy

# 方法三：从源码安装
git clone https://github.com/sympy/sympy.git
cd sympy
pip install .

>>> import sympy
>>> sympy.__version__
'1.14.0'

# 启动 SymPy 控制台（增强版Python交互环境）
$ bin/isympy

# 或直接使用 isympy 命令
$ isympy

>>> from sympy import Symbol, cos, sin, exp, sqrt
>>> from sympy import integrate, differentiate, limit, series

# 定义符号
>>> x = Symbol('x')
>>> y = Symbol('y')

# 表达式创建
>>> e = (x + y)**2
>>> e
(x + y)**2

# 展开
>>> e.expand()
x**2 + 2*x*y + y**2

# 三角函数化简
>>> cos(x)**2 + sin(x)**2
1

>>> x = Symbol('x')

# 求导
>>> diff(x**3, x)
3*x**2

# 不定积分
>>> integrate(x**2, x)
x**3/3

# 定积分
>>> integrate(0, (x, 0, 1))
1

# 极限
>>> limit(1/x, x, 0, dir='+')
oo

# 泰勒展开
>>> series(exp(x), x, 0, 10)
1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + x**6/720 + x**7/5040 + x**8/40320 + x**9/362880 + O(x**10)

>>> from sympy import solve, solveset

>>> x = Symbol('x')

# 代数方程求解
>>> solve(x**2 - 4, x)
[-2, 2]

# 微分方程求解
>>> f = Function('f')
>>> x = Symbol('x')
>>> dsolve(f(x).diff(x) - f(x), f(x))
f(x) == C1*e**x

# 求解方程组
>>> solve([x + y - 2, x - y - 0], [x, y])
{x: 1, y: 1}

>>> from sympy import Matrix

>>> M = Matrix([[1, 2], [3, 4]])
>>> M
Matrix([
[1, 2],
[3, 4]])

# 行列式
>>> M.det()
-2

# 逆矩阵
>>> M.inv()
Matrix([
[ -2,   1],
[3/2, -1/2]])

# 特征值
>>> M.eigenvals()
{-2: 1, 3: 1}

# 矩阵乘法
>>> M * M
Matrix([
[ 7, 10],
[15, 22]])

>>> from sympy import latex, Symbol
>>> x = Symbol('x')
>>> print(latex(x**2 + cos(x)))
x^{2} + \cos\left(x\right)

>>> # 积分
>>> from sympy import Integral
>>> print(latex(Integral(x**2, x)))
\int x^{2}\,dx

>>> from sympy import lambdify
>>> x = Symbol('x')
>>> expr = x**2 + 2*x + 1

# 转换为NumPy函数
>>> f = lambdify(x, expr, 'numpy')
>>> f(3)
16

# 使用mpmath高精度
>>> import mpmath as mp
>>> f_mp = lambdify(x, expr, 'mpmath')
>>> f_mp(mp.mpf('0.5'))
2.25

>>> from sympy import Matrix, symbols, Function

>>> a, b, c = symbols('a b c')
>>> M = Matrix([[a, b], [c, a]])
>>> M
Matrix([
[a, b],
[c, a]])

>>> M.charpoly()
PurePoly(lambda**2 - 2*a*lambda + a**2 - b*c, lambda, domain='ZZ[a,b,c]')

>>> M.eigenvects()
[(a - sqrt(b*c), 1, [Matrix([
[-b/(a - sqrt(b*c))],
[                     1]])],
 (a + sqrt(b*c), 1, [Matrix([
[-b/(a + sqrt(b*c))],
[                     1]]))]

>>> from sympy import isprime, prime, factorint, totient

# 素数测试
>>> isprime(97)
True

# 第100个素数
>>> prime(100)
547

# 整数分解
>>> factorint(123456)
{2: 6, 3: 1, 643: 1}

# 欧拉函数
>>> totient(100)
40

>>> from sympy.physics.mechanics import *
>>> from sympy import symbols, Function

>>> q1, q2 = dynamicsymbols('q1 q2')
>>> q1_d, q2_d = dynamicsymbols('q1 q2', 1)

>>> M = Matrix([[2, 0], [0, 2]])
>>> F = Matrix([[2*q2_d**2], [0]])

>>> M
Matrix([
[2, 0],
[0, 2]])

>>> F
Matrix([
[2*q2_d**2],
[      0]])

>>> from sympy.physics.quantum import *
>>> from sympy import symbols, I, hbar

>>> a = AnnihilationOperator('a')
>>> a_d = CreationOperator('a')

>>> a * a_d  # 不对易
I + a_d*a

>>> a_d * a  # 粒子数算符
a_d*a

>>> from sympy import symbols, plot, sin, cos, exp
>>> x = symbols('x')

# 基础绘图
>>> p1 = plot(sin(x), (x, -pi, pi))

# 多函数绘图
>>> p2 = plot(sin(x), cos(x), exp(-x), (x, -2*pi, 2*pi))

# 保存
>>> p2.save('plot.png')

>>> from sympy import symbols, plot3d
>>> x, y = symbols('x y')

>>> plot3d(x**2 + y**2, (x, -5, 5), (y, -5, 5))

>>> from sympy import simplify, trigsimp

# 三角简化
>>> trigsimp(sin(x)**2 + cos(x)**2)
1

# 通用简化
>>> simplify((x**2 + 2*x + 1)/(x + 1))
x + 1

>>> from sympy import N, sqrt, pi

# 数值计算
>>> N(sqrt(2))
1.41421356237310

>>> N(pi, dps=50)  # 50位精度
3.1415926535897932384626433832795028841971693993751

>>> import numpy as np
>>> from sympy import lambdify

>>> x = Symbol('x')
>>> expr = sin(x) + cos(x)

>>> # 转NumPy函数
>>> f = lambdify(x, expr, 'numpy')
>>> f(np.array([0, pi/2, pi]))
array([ 1.        ,  1.41421356,  -1.        ])

>>> import matplotlib.pyplot as plt
>>> from sympy import plot

>>> p = plot(sin(x), (x, 0, 2*pi), show=False)
>>> p.xlabel = 'x'
>>> p.ylabel = 'sin(x)'
>>> p.show()

>>> from sympy import latex, Integral, Symbol

>>> x = Symbol('x')
>>> expr = Integral(x**2, (x, 0, 1))

>>> print(latex(expr))
\int_{0}^{1} x^{2}\,dx

>>> # 在Jupyter中显示
>>> expr  # 自动渲染为LaTeX

# 克隆仓库
git clone https://github.com/sympy/sympy.git
cd sympy

# 安装开发依赖
pip install -r requirements-dev.txt

# 以开发模式安装
pip install -e .

# 运行测试
./setup.py test

# 运行特定测试
bin/test sympy/core/tests/test_arit.py

@article{meurer2017sympy,
  title={SymPy: symbolic computing in Python},
  author={Meurer, Aaron and Smith, Christopher P and Paprocki, Mateusz
           and {\v{C}}ert{\'i}k, Ond{\v{r}}ej and Kirpichev, Sergey B
           and Rocklin, Matthew and Kumar, Amit and Ivanov, Sergiu
           and Moore, Jason K and Singh, Sartaj and Rathnayake, Thilina
           and Vig, Sean and Granger, Brian E and Muller, Richard P
           and Bonazzi, Francesco and Gupta, Harsh and Vats, Shivam
           and Johansson, Fredrik and Pedregosa, Fabian and Curry, Matthew J
           and Terrel, Andy R and Rou{\v{c}}ka, {\v{S}}t{\v{e}}p{\'a}n
           and Saboo, Ashutosh and Fernando, Isuru and Kulal, Sumith
           and Cimrman, Robert and Scopatz, Anthony},
  journal={PeerJ Computer Science},
  volume={3},
  pages={e103},
  year={2017}
}