Python solves advanced mathematics problems, and my mother no longer has to worry about my learning
Use Python to solve limits and derivatives in advanced mathematics , partial derivatives, definite integrals, indefinite integrals, double integrals and other problems
Sympy is a Python scientific computing library, which aims to become a fully functional computer algebra system. SymPy includes functions from basic symbolic arithmetic to calculus, algebra, discrete mathematics and quantum physics. It can display the results in LaTeX.
Sympy official website
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- Python solves advanced mathematics problems, my mother no longer has to worry about my study
- 1. Practical skills
- 1.1 Symbolic function
- 1.2 Expansion expression expand
- 1.3 Taylor expansion formula series
- 1.4 Sign expansion
- 2. Find the limit limit
- 3. Find the derivative diff
- 3.1 Unary function
- 3.2 Multivariate function
- 4. Integral integrate
- 4.1 Definite integral
- 4.2 Indefinite integral
- 4.3 Double integral
- 5. Solve the system of equations solve
- 6. Calculate the summation
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from sympy import *import sympy
Enter the "x= symbols("x")" command to define a symbol
x = Symbol("x")y = Symbol("y")
1. Practical tips
1.1 Symbol function
sympy provides a lot of mathematical symbols, summarized as follows
- Imaginary unit
sympy.I
- Natural logarithm
sympy.E
- Infinity
sympy.oo
- Pi
sympy.pi
- Find the nth square root
sympy.root(8,3)
- Get the logarithm
sympy.log(1024,2)
- Find the factorial
sympy.factorial(4)
- Trigonometric function
sympy.sin(sympy.pi)sympy.tan(sympy.pi/4)sympy.cos(sympy.pi/2)
1.2 Expand expression expand
f = (1+x)**3expand(f)
x 3 + 3 x 2 + 3 x + 1 \displaystyle x^{3} + 3 x^{2} + 3 x + 1 x3+3x2+3x+1
1.3 泰勒展开公式series
ln(1+x).series(x,0,4)
x − x 2 2 + x 3 3 + O ( x 4 ) \displaystyle x - \frac{x^{2}}{2} + \frac{x^{3}}{3} + O\left(x^{4}\right) x−2x2 3x3 O(x4)
sin(x).series(x,0,8)
x − x 3 6 + x 5 120 − x 7 5040 + O ( x 8 ) \displaystyle x - \frac{x^{3}}{6} + \frac{x^{5}}{120} - \frac{x^{7}}{5040} + O\left(x^{8}\right) x−6x3 120x5−5040x7 O(x8)
cos(x).series(x,0,9)
1 − x 2 2 + x 4 24 − x 6 720 + x 8 40320 + O ( x 9 ) \displaystyle 1 - \frac{x^{2}}{2} + \frac{x^{4}}{24} - \frac{x^{6}}{720} + \frac{x^{8}}{40320} + O\left(x^{9}\right) 1−2x2 24x4−720x6 40320x8 O(x9)
(1/(1+x)).series(x,0,5)
1 − x + x 2 − x 3 + x 4 + O ( x 5 ) \displaystyle 1 - x + x^{2} - x^{3} + x^{4} + O\left(x^{5}\right) 1−x x2−x3 x4 O(x5)
tan(x).series(x,0,4)
x + x 3 3 + O ( x 4 ) \displaystyle x + \frac{x^{3}}{3} + O\left(x^{4}\right) x+3x3+O(x4)
(1/(1-x)).series(x,0,4)
1 + x + x 2 + x 3 + O ( x 4 ) \displaystyle 1 + x + x^{2} + x^{3} + O\left(x^{4}\right) 1+x+x2+x3+O(x4)
(1/(1+x)).series(x,0,4)
1 − x + x 2 − x 3 + O ( x 4 ) \displaystyle 1 - x + x^{2} - x^{3} + O\left(x^{4}\right) 1−x+x2−x3+O(x4)
1.4 符号展开
a = Symbol("a")b = Symbol("b")#simplify( )普通的化简simplify((x**3 + x**2 - x - 1)/(x**2 + 2*x + 1))#trigsimp( )三角化简trigsimp(sin(x)/cos(x))#powsimp( )指数化简powsimp(x**a*x**b)
x a + b \displaystyle x^{a + b} xa+b
2. 求极限limit
limit(sin(x)/x,x,0)
1 \displaystyle 1 1
f2=(1+x)**(1/x)
f2
( x + 1 ) 1 x \displaystyle \left(x + 1\right)^{\frac{1}{x}} (x+1)x1
重要极限
f1=sin(x)/x f2=(1+x)**(1/x)f3=(1+1/x)**x lim1=limit(f1,x,0)lim2=limit(f2,x,0)lim3=limit(f3,x,oo)print(lim1,lim2,lim3)
1 E E
dir可以表示极限的趋近方向
f4 = (1+exp(1/x))f4
e 1 x + 1 \displaystyle e^{\frac{1}{x}} + 1 ex1+1
lim4 = limit(f4,x,0,dir="-")lim4
1 \displaystyle 1 1
lim5 = limit(f4,x,0,dir="+")lim5
∞ \displaystyle \infty ∞
3. 求导diff
diff(函数,自变量,求导次数)
3.1 一元函数
求导问题
diff(sin(2*x),x)
2 cos ( 2 x ) \displaystyle 2 \cos{\left(2 x \right)} 2cos(2x)
diff(ln(x),x)
1 x \displaystyle \frac{1}{x} x1
3.2 多元函数
求偏导问题
diff(sin(x*y),x,y)
− x y sin ( x y ) + cos ( x y ) \displaystyle - x y \sin{\left(x y \right)} + \cos{\left(x y \right)} −xysin(xy)+cos(xy)
4. 积分integrate
4.1 定积分
- 函数的定积分: integrate(函数,(变量,下限,上限))
- 函数的不定积分: integrate(函数,变量)
f = x**2 + 1integrate(f,(x,-1.1))
− 1.54366666666667 \displaystyle -1.54366666666667 −1.54366666666667
integrate(exp(x),(x,-oo,0))
1 \displaystyle 1 1
4.2 不定积分
f = 1/(1+x*x)integrate(f,x)
atan ( x ) \displaystyle \operatorname{atan}{\left(x \right)} atan(x)
4.3 双重积分
f = (4/3)*x + 2*y integrate(f,(x,0,1),(y,-3,4))
11.6666666666667 \displaystyle 11.6666666666667 11.6666666666667
5. 求解方程组solve
#解方程组#定义变量f1=x+y-3f2=x-y+5solve([f1,f2],[x,y])
{x: -1, y: 4}
6. 计算求和式summation
计算求和式可以使用sympy.summation函数,其函数原型为sympy.summation(f, *symbols, **kwargs)
**
sympy.summation(2 * n,(n,1,100))
10100
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