python - SymPy

pip install sympy

import sympy

# 虚数单位i
sympy.I  # 𝑖
sympy.I ** 2  # -1
sympy.sqrt(-1)  # 𝑖

# 自然对数的底e
sympy.E  # e
sympy.log(sympy.E)  # 1

# 无穷大
1/sympy.oo  # 0
1+sympy.oo  # oo

# 圆周率pi
sympy.pi  # 𝜋
sympy.sin(sympy.pi/2)  # 1

初等运算

# 求对数
sympy.log(sympy.E)  # 1
sympy.log(100, 10)  # 2

# 求平方根
sympy.sqrt(4)  # 2
sympy.sqrt(-1)  # 𝑖

# 求n次方根
sympy.root(8,3)  # 2

# 求k次方
2**3  # 8
16**(1/2)  # 4.0

# 求阶层
sympy.factorial(20)  # 2432902008176640000

# 求三角函数
sympy.sin(sympy.pi)  # 0
sympy.cos(sympy.pi)  # -1

# 表达式与表达式求值

表达式与表达式求值

# 定义x为一个符号，表示为一个变量
x = sympy.Symbol('x')
f = 2*x + 1
# 使用evalf函数，传入变量x的值，对表达式进行求值
f.evalf(subs={x:4})  # 9.0

x, y = sympy.symbols('x y')
f = 2*x + y
f.evalf(subs={x:4,y:2})  # 10.0
f.evalf(subs={x:4})  # 𝑦+8.0

解方程

x = sympy.Symbol('x')

# 解方程 x - 2 = 0
sympy.solve(x - 2, x)  # [2]

# 解方程 x^2 - 1 = 0
sympy.solve(x**2 - 1, x)  # [-1, 1]

# 解方程 x^2 + 1 = 0
sympy.solve(x**2 + 1, x)  # [-I, I]

f = x + 1
sympy.solve(f, x)  # -1

# 解方程组
# x + y = 1
# x - y = 3
x, y = sympy.symbols('x y')
f1 = x + y - 1
f2 = x - y - 3
sympy.solve([f1, f2], [x, y])  # {x: 2, y: -1}

计算求和式

${\displaystyle \sum _{n=1}^{100}2n}$

n = sympy.Symbol('n')
sympy.summation(2 * n, (n, 1, 100))  # 10100

求和式的方程

${\displaystyle \sum _{i=1}^{5}x+10x=15}$

x = sympy.Symbol('x')
i = sympy.Symbol('i',integer = True)
f =  sympy.summation(x,(i,1,5)) + 10 * x - 15
sympy.solve(f,x)  # [1]

求极限

${\displaystyle \underset{x\to 0}{lim}\frac{x}{sinx}=1}$

x = sympy.Symbol('x')
f = sympy.sin(x) / x
sympy.limit(f, x, 0)  # 1

${\displaystyle \underset{x\to 0}{lim}(1+x{)}^{\frac{1}{x}}=e}$

x = sympy.Symbol('x')
f = (1+x)**(1/x)
sympy.limit(f, x, 0)  # e

${\displaystyle \underset{x\to +\mathrm{\infty }}{lim}(1+\frac{1}{x}{)}^x=e}$

x = sympy.Symbol('x')
f = (1+1/x)**x
sympy.limit(f, x, sympy.oo)  # e

求导

x = sympy.Symbol('x')
f = x ** 2 + 2 * x + 1
sympy.diff(f,x)  # 2𝑥+2

x = sympy.Symbol('x')
f = sympy.sin(x)
sympy.diff(f,x)  # cos(𝑥)

x, y = sympy.symbols('x y')
f = x**2 + 2*x + y**3
sympy.diff(f,x)  # 2𝑥+2
sympy.diff(f,y)  # 3𝑦^2

定积分

${\int }_0^{1}2xdx$

x = sympy.Symbol('x')
f = 2 * x
sympy.integrate(f,(x,0,1))  # 1

${\int }_0^3{\int }_0^{x}2xdxdx$

x = sympy.Symbol('x')
f = 2 * x
g = sympy.integrate(f,(x,0,x))  # x^2
sympy.integrate(g,(x,0,3))  # 9

不定积分

$\int (e^x+2x)dx$

x = sympy.Symbol('x')
f = sympy.E**x + 2*x
sympy.integrate(f,x)  # x^2 + e^2