Here I'd like to share how to use a Python library "SymPy", which is for symbolic formula manipulation.
- How to use SymPy
- Import SymPy library
- Addition, Subtraction, Multiplication and Division
- Modulo
- Type of values
- Power, Factorial, Double factorial, Square root
- Complex numbers
- Mathematical constants
- Refer numbers
- Algebraic manipulations
- Declare the properties of variables
- Summation
- Multiple products
- Elementary functions
- Manipulations over trigonometric/hyperbolic functions
- Equation
- Function
- Limit / Improper integral
- Matrices
- Ordinary differential equations
- Fourier analysis
- Lists
- FOR
- Display in TeX form
- Output in TeX form
- Output in SymPy source
- Initialize a letter
- Initialize all vriables
How to use SymPy
In the following I'll show you concrete examples of input of Python (SymPy). Although I've checked them on JupyterNotebook, please let me know if you found any trouble. Copy, Paste and Run it (Shift + Enter)! Modify the input as you like! Let's get it started!
Import SymPy library
First of all, one must load SymPy library:
import sympy
Addition, Subtraction, Multiplication and Division
3 + 2 # letters after "#" is to be ignored for evaluation
3 - 2
3 * 2
sympy.Rational(3,2)
or
3 / 2
Modulo
11 % 2
11 // 2
Type of values
type( 3 )
type( 3. )
Power, Factorial, Double factorial, Square root
3 ** 2
3e2
3e-2
sympy.factorial(10)
sympy.factorial2(10)
sympy.factorial2(9)
sympy.sqrt(50)
Complex numbers
Imaginary unit
sympy.I**2 # capital " I "
Complex conjugate
sympy.conjugate(2+3*sympy.I)
Real part, Imaginary part
sympy.re(2+sympy.I*3)
sympy.im(2+sympy.I*3)
Absolute value
sympy.Abs(2+sympy.I*3)
Argument
sympy.arg(2+sympy.I*3)
Mathematical constants
Pi
sympy.pi
Napier number
sympy.E
Euler's formula
sympy.E**( sympy.pi * sympy.I )
Infinity
sympy.oo
Refer numbers
sympy.pi.evalf(5) # argument specifies how many digits you want to show
Algebraic manipulations
Prime factorization
sympy.factorint(sympy.factorial(10))
Declare variables
sympy.var('x')
or
x = sympy.symbols('x')
Factorization of polynomial
sympy.factor(x**2-4*x+3)
Partial fraction decomposition
sympy.apart(1/(x**2-4*x+3))
sympy.apart(1/(x**6-1))
Expand expression
sympy.var('x, y') sympy.expand((x+y)**6)
Simplify expression
sympy.simplify((2*x-2)/((x-1)**2*(x-2)))
sympy.simplify(1/(x-1)**2+1/((x-1)*(x-2)))
Simplify and expand expression
sympy.cancel((2*x-2)/((x-1)**2*(x-2)))
sympy.cancel(1/(x-1)**2+1/((x-1)*(x-2)))
Examine True or False
0==sympy.sqrt(2)-2**(sympy.Rational(1,2))
0==sympy.sqrt(2)-2**(sympy.Rational(1,3))
sympy.factorial(50)==sympy.factorial2(50)*sympy.factorial2(49)
sometimes it needs some manipulation on expressions
x**2-2*x+1==(x-1)**2
x**2-2*x+1==sympy.expand((x-1)**2)
Extract a coefficient
( x**2-2*x+1 ).coeff(x, 2)
Substitution
sympy.var('x, a, b, c') (a*x**2+b*x+c).subs(x, 3)
sympy.expand((a*x**2+b*x+c).subs(x, (-b + sympy.sqrt(-4*a*c + b**2))/(2*a)))
Declare the properties of variables
Integer
k = sympy.symbols('k',integer=True)
Real number
a, b = sympy.symbols('a, b', real=True)
sympy.conjugate(a+b*sympy.I)
unless declaring anything, SymPy regards variables as complex:
sympy.var('z')
sympy.conjugate(z)
Non-negative
sympy.var('r', nonnegative = True)
sympy.sqrt(r**2)
Positive
sympy.var('t', positive = True)
sympy.Abs(t)
Summation
k = sympy.symbols('k',integer=True) sympy.summation(k, (k, 1, 10) )
k, N = sympy.symbols('k N',integer=True) sympy.factor(sympy.summation(k, (k, 1, N) ))
sympy.factor(sympy.summation(k**3, (k, 1, N) ))
Multiple products
k = sympy.symbols('k',integer=True) sympy.product( k, (k, 1, 10) )
sympy.factorial(10)==sympy.product(k, (k, 1, 10) )
Elementary functions
Trigonometric functions
sympy.sin(x)
sympy.cos(x)
sympy.tan(x)
sympy.sec(x)
sympy.csc(x)
sympy.cot(x)
Inverse trigonometric functions
sympy.asin(x)
sympy.acos(x)
sympy.atan(x)
Hyperbolic functions
sympy.sinh(x)
sympy.cosh(x)
sympy.tanh(x)
Exponential function
sympy.exp(x)
sympy.var('theta', real = True) sympy.re(sympy.exp(sympy.I*theta))
Logarithmic function
sympy.log(x)
Manipulations over trigonometric/hyperbolic functions
Expansions
sympy.var('a, b') sympy.expand(sympy.sin(a+b), trig=True)
Simplifications
sympy.trigsimp((sympy.sinh(a))**2+(sympy.cosh(a))**2)
Equation
Solve simultaneous equations
sympy.var('x, y') sympy.solve ([3*x+4*y-5, 2*x+3*y-3], [x, y])
Find roots of algebraic equation
sympy.var('x, a, b, c') sympy.solve (a*x**2+b*x+c, x)
For more details, see
Function
Assign expressions to a letter, and evaluate it at a value
sympy.var('x, y') u=x+y v=(x+y)**3 u.subs(x,5)
(v.subs(x,5)).subs(y,3)
Use assigned letter: 4 operations
u+v
sympy.expand(u+v)
sympy.factor(u+v)
sympy.expand(u*v)
sympy.simplify(u/v)
Use assigned letter: differentiation
sympy.diff(v, x)
sympy.diff(v, y, 2)
or
v.diff(y, 2)
0==sympy.diff(v, x, 1)-sympy.diff(v, x)
Use assigned letter: Integration
sympy.integrate(u, x)
sympy.integrate(v, (x, 0, 6))
Define a function
sympy.var('x') def f(x): return sympy.log(1+x)/x f(1)
Differentiation
sympy.diff(f(x), x, 1)
Integration
sympy.integrate(f(x), x)
Taylor expansion
sympy.series(f(x), x, 0, 7)
Extract RHS, LHS, Denominator, Numerator
sympy.var('x') F=x+1 G=x**2+x-2 E=sympy.Eq(x+1, F/G) sympy.solve(E, x)
E.rhs
E.lhs
sympy.denom(E.rhs)
sympy.numer(E.rhs)
Limit / Improper integral
Limit
sympy.limit( sympy.sin(x)/x, x, 0 )
Improper integral
sympy.integrate(1/(sympy.sqrt(sympy.pi))*sympy.exp(-x**2), (x, -sympy.oo, sympy.oo) )
Matrices
Input matrices
sympy.var('a11, a12, a21, a22, b11, b12, b21, b22')
A = sympy.Matrix([
[a11,a12],
[a21,a22]])
B = sympy.Matrix([
[b11,b12],
[b21,b22]])
Refer the element
A[0,1]
Linear combination
sympy.var('c, d')
c * A + d * B
Matrix products
A * B
Power of matrices
A ** 3
Determinant
A.det()
Ordinary differential equations
sympy.var('a, x') y = sympy.Function('y')(x) eq = sympy.Eq( sympy.diff(y, x), -a*y ) sympy.dsolve(eq)
For more details, see
Fourier analysis
Lists
Make List, Refer its element
S=list([1, 10, 100]) S[0]
S[0]+S[1]*S[2]
S[3]
Summation over all elements in a list
sum(S)
Min/Max elements in a list
min(S)
max(S)
Number of elements in a list
len(S)
FOR
def f(k): return k**2 List = [] for n in range(10): element = f(n) List.append(element) print(List)
Display in TeX form
sympy.var('a, b, c, x') E2=sympy.Eq(a*x**2+b*x+c, 0) sol=sympy.solve(E2, x) display(sol[1])
sympy.init_printing()
display(sol[1])
Output in TeX form
sympy.var('a, b, c, x') E2=sympy.Eq(a*x**2+b*x+c, 0) sol=sympy.solve(E2, x) display(sol[1])
print(sympy.latex(sol[0]))
Output in SymPy source
sympy.var('x, y') exp1 = sympy.log((1+x)*(1+y)**2) exp2 = sympy.expand(exp1) sympy.print_python( exp2 )
Initialize a letter
del(x)
Initialize all vriables
%reset
As we've seen so far, many objects in SymPy are associated with " sympy. "
we can shorten it by doing from the beginning
from sympy import sp
then, for example
sp.I**2
Keywords: Python, SymPy, JupyterNotbook
