Basic Operations
expr.subs([(x, 2), (y, 4), (z, 0)]) |
substitute x with 2 etc. |
sympify(str_expr) |
convert strings into SymPy expressions |
expr.evalf(15, chop=True) |
evaluate a numerical expression into a floating point number |
lambdify(x, expr, ”numpy”) |
converts the SymPy names to the names of the given numerical library |
init_printing() |
This will automatically enable the best printer available in your environment. |
simplify(expr) |
simplify mathematical expressions |
expand(expr) |
expand polynomial expressions |
factor(expr) |
takes a polynomial and factors it into irreducible factors over the rational numbers |
factor_list(expr) |
returns a list with the factors. More structured. |
collect(expr, x) |
collects common powers of a term in an expression |
cancel(expr) |
take any rational function and put it into the standard canonical form |
apart(expr) |
performs a partial fraction decomposition on a rational function |
Matrices
Matrix([1, 2, 3]) |
matrix constructor(mutable matrix) |
shape(expr) |
shape of matrix |
M.row(0) |
get the first row |
M.col(-1) |
get the last column |
M.col_del(0) |
delete first column |
M.row_del(1) |
delete second row |
M.row_insert(1, Matrix([[0, 4]])) |
insert a row |
M.col_insert(0, Matrix([1, -2])) |
insert a column |
M**-1 |
inverse of M |
M.T |
transpose of M |
eye(n) |
create a nxn identity matrix |
zeros(n,m) |
creates a nxm matrix of zeroes |
ones(n,m) |
creates a nxm matrix of ones |
diag(expr) |
creates a matrix with expr in the diagonal |
M.det() |
computes the determinant of M |
M.rref() |
put a matrix into reduced row echelon form |
M.nullspace() |
returns a list of column vectors that span the nullspace of the matrix |
M.columnspace() |
returns a list of column vectors that span the columnspace of the matrix |
M.eigenvals() |
eigenvals returns a dictionary of eigenvalue: algebraic_multiplicity pairs |
M.eigenvects() |
returns a list of tuples of the form (eigenvalue, algebraic_multiplicity, [eigenvectors]) |
M.diagonalize() |
returns a tuple (P, D), where D is diagonal and M = P DP **−1 |
M.charpoly(lamda) |
return the characteristic polynomial |
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Trigonometric Simplification
trigsimp(expr) |
simplify expressions using trigonometric identities |
expand_trig(expr) |
expand trigonometric functions |
Powers
powsimp(expr) |
use power identities |
expand_power_exp(x**(a + b)) |
x**a * x**b |
expand_power_base((xy)*a) |
x**a * y**a |
powdenest((xa)b)powdenest((xa)b) |
x**(a*b) |
Exponentials and logarithms
expand_log(expr) |
logcombine(expr) |
Special Functions
factorial(n) |
return the factorial of n |
binomial(n, k) |
return the binomial coefficient of n and k |
gamma(z) |
return the gamma function |
expr.rewrite(function) |
rewrite expr in terms of function |
expand_func(expr) |
expand special functions |
hyperexpand(expr) |
rewrite hyper in terms of more standard functions |
combsimp(expr) |
simplify combinatorial expressions |
gammasimp(expr) |
simplify expressions with gamma functions or combinatorial functions |
Assumptions
positive |
negative |
real |
complex |
integer |
expr.assumptions0 |
The full set of known predicates for a symbol |
posify(expr) |
replace all symbols in an expression with symbols that have the assumption positive=True |
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Calculus
diff(expr, x, n) |
nth order derivative of expr in terms of x |
Derivative(expr, x, n) |
create an unevaluated derivative |
deriv.doit() |
evaluate an unevaluated derivative |
integrate(expr, x, a, b) |
integrate expr from a to b |
Integral(expr, x, n) |
create an unevaluated integral |
limit(expr, x, xo) |
limit of expr to xo |
Limit(expr, x, xo) |
create an unevaluated limit |
expr.series(x, x0, n) |
nth order series expansion of expr around x0 |
expr.series(x, x0, n).removeO() |
remove O notation |
differentiate_finite(expr) |
differentiate using finite differences |
expr.as_finite_difference() |
generate approximations of the derivative to arbitrary order |
Solvers
solveset(expr, x, domain=S.Complexes, dict=False) |
solve expr=0 |
linsolve([expr1, expr2, ...], (x, y, ...)) |
solve a linear system of equations |
nonlinsolve([expr1, expr2, ...], [x, y, ...]) |
solve a non linear system of equations |
dsolve(diffeq, f(x)) |
soves differential equation diffeq |
roots(expr, x) |
o get the solutions of a polynomial including multiplicity |
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