sympy.md

SymPy

Version: 1.14.0 | Type: Stock (pure Python) | Status: Fully working

SymPy is a pure Python symbolic math library -- no compiled extensions, runs natively on iOS without patches.

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Quick Start

from sympy import symbols, solve, diff, integrate, sin, cos, exp, pi, oo, series, limit, simplify, factor, expand, Eq, sqrt, Rational, Matrix

x, y, z = symbols('x y z')

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Core Module -- sympy.core

Symbols & Constants

Object	Description
symbols('x y z')	Create symbolic variables
Symbol('x', positive=True)	Symbol with assumptions
Rational(p, q)	Exact rational number p/q
Integer(n)	Symbolic integer
Float(x, dps)	Symbolic float with precision
pi	3.14159...
E	2.71828... (Euler's number)
I	Imaginary unit
oo	Positive infinity
zoo	Complex infinity
nan	Not a number
S.Half / S.One / S.Zero	Singleton rationals
S.NegativeOne	-1
S.Infinity	Same as oo
GoldenRatio	(1 + sqrt(5))/2
EulerGamma	Euler-Mascheroni constant
Catalan	Catalan's constant

Expression Manipulation

Function	Description
simplify(expr)	General simplification
expand(expr)	Expand products and powers
factor(expr)	Factor polynomial
collect(expr, x)	Collect coefficients of x
cancel(expr)	Cancel common factors
apart(expr, x)	Partial fraction decomposition
together(expr)	Combine fractions
radsimp(expr)	Rationalize denominator
powsimp(expr)	Simplify powers
trigsimp(expr)	Simplify trig expressions
logcombine(expr)	Combine logarithms
nsimplify(expr)	Convert float to exact
cse(exprs)	Common subexpression elimination
subs(old, new)	Substitute values
expr.evalf(n)	Numerical evaluation to n digits
N(expr, n)	Same as evalf
expr.rewrite(func)	Rewrite in terms of func
expr.as_numer_denom()	Split to numerator/denominator
expr.coeff(x, n)	Coefficient of x^n
degree(expr, x)	Degree of polynomial
Poly(expr, x)	Convert to polynomial object

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Solvers -- sympy.solvers

Function	Description
solve(expr, x)	Solve equation(s) symbolically
solve([eq1, eq2], [x, y])	Solve system of equations
solve(Eq(lhs, rhs), x)	Solve explicit equation
solveset(expr, x, domain)	Solve with domain specification (Reals, Complexes)
linsolve([eq1, eq2], x, y)	Solve linear system (returns set)
nonlinsolve([eq1, eq2], [x, y])	Solve nonlinear system
nsolve(expr, x0)	Numerical root finding
roots(expr, x)	Roots with multiplicity
real_roots(expr, x)	Real roots only
dsolve(ode, f(x))	Solve ordinary differential equation
pdsolve(pde, f(x, y))	Solve partial differential equation
checkodesol(ode, sol)	Verify ODE solution
reduce_inequalities(ineqs, x)	Solve inequalities
diophantine(expr)	Solve Diophantine equations

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Calculus -- sympy.calculus

Function	Description
diff(expr, x)	First derivative d/dx
diff(expr, x, n)	N-th derivative
diff(expr, x, y)	Mixed partial derivative
Derivative(expr, x)	Unevaluated derivative
integrate(expr, x)	Indefinite integral
integrate(expr, (x, a, b))	Definite integral from a to b
Integral(expr, x)	Unevaluated integral
limit(expr, x, x0)	Limit as x -> x0
limit(expr, x, x0, '+')	Right-hand limit
limit(expr, x, x0, '-')	Left-hand limit
series(expr, x, x0, n)	Taylor/Laurent series around x0
summation(expr, (i, a, b))	Symbolic sum
product(expr, (i, a, b))	Symbolic product
sequence(expr, (n, a, b))	Define sequence
fourier_series(f, (x, -pi, pi))	Fourier series expansion
singularities(expr, x)	Find singularities
is_increasing(expr, interval)	Test if increasing
is_decreasing(expr, interval)	Test if decreasing
minimum(expr, x, domain)	Find minimum
maximum(expr, x, domain)	Find maximum

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Matrices -- sympy.matrices

Function / Class	Description
Matrix([[a, b], [c, d]])	Create matrix
eye(n)	Identity matrix
zeros(m, n)	Zero matrix
ones(m, n)	Ones matrix
diag(*args)	Diagonal matrix
M.det()	Determinant
M.inv()	Inverse
M.transpose() / M.T	Transpose
M.adjugate()	Adjugate (classical adjoint)
M.cofactor(i, j)	Cofactor
M.eigenvals()	Eigenvalues with multiplicities
M.eigenvects()	Eigenvalues and eigenvectors
M.diagonalize()	Diagonalization (P, D)
M.jordan_form()	Jordan normal form
M.rref()	Reduced row echelon form
M.rank()	Matrix rank
M.nullspace()	Null space basis
M.columnspace()	Column space basis
M.rowspace()	Row space basis
M.norm(ord)	Matrix norm
M.trace()	Trace
M.cholesky()	Cholesky decomposition
M.LUdecomposition()	LU decomposition
M.QRdecomposition()	QR decomposition
M.singular_values()	Singular values
M.condition_number()	Condition number
M.exp()	Matrix exponential
M.applyfunc(f)	Apply function to each element
M.row_del(i) / M.col_del(j)	Delete row/column
M.row_insert(i, row)	Insert row
M.col_insert(j, col)	Insert column
M * N	Matrix multiplication
M ** n	Matrix power
M.cross(N)	Cross product (3D vectors)
M.dot(N)	Dot product

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Functions -- sympy.functions

Trigonometric

sin, cos, tan, cot, sec, csc, asin, acos, atan, acot, asec, acsc, atan2

Hyperbolic

sinh, cosh, tanh, coth, sech, csch, asinh, acosh, atanh, acoth

Exponential & Logarithmic

exp, log, ln, LambertW

Power & Roots

sqrt, cbrt, root(x, n), Pow, Abs, sign

Combinatorial

factorial, binomial, fibonacci, lucas, harmonic, bernoulli, euler, catalan, bell, stirling, subfactorial, RisingFactorial, FallingFactorial

Special Functions

gamma, loggamma, digamma, polygamma, beta, zeta, dirichlet_eta, polylog, lerchphi, Ei, Si, Ci, li, erf, erfc, erfi, erfinv, erfcinv, besselj, bessely, besseli, besselk, hankel1, hankel2, jn, yn, airyai, airybi, legendreP, legendreQ, assoc_legendre, hermite, laguerre, assoc_laguerre, chebyshevt, chebyshevu, jacobi, gegenbauer, Ynm (spherical harmonics), hyper, meijerg, elliptic_k, elliptic_e, elliptic_pi

Piecewise & Other

Piecewise((expr1, cond1), (expr2, cond2), ...), Heaviside(x), DiracDelta(x), floor, ceiling, frac, Min, Max, re, im, conjugate, arg

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Number Theory -- sympy.ntheory

Function	Description
isprime(n)	Primality test
nextprime(n)	Next prime after n
prevprime(n)	Previous prime before n
prime(n)	N-th prime
primerange(a, b)	Primes in range [a, b)
primepi(n)	Count of primes <= n
factorint(n)	Integer factorization (dict)
divisors(n)	All divisors
divisor_count(n)	Number of divisors
divisor_sigma(n, k)	Sum of k-th powers of divisors
totient(n)	Euler's totient function
reduced_totient(n)	Carmichael's lambda
mobius(n)	Mobius function
gcd(a, b) / lcm(a, b)	GCD / LCM
igcd(a, b)	Integer GCD
mod_inverse(a, m)	Modular inverse
is_quad_residue(a, p)	Quadratic residue test
legendre_symbol(a, p)	Legendre symbol
jacobi_symbol(a, n)	Jacobi symbol
discrete_log(n, a, b)	Discrete logarithm
continued_fraction_periodic(p, q, d)	Periodic continued fraction
egyptian_fraction(r)	Egyptian fraction representation
binomial_coefficients(n)	All binomial coefficients of n
npartitions(n)	Number of partitions

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Geometry -- sympy.geometry

Class	Description
Point(x, y) / Point3D(x, y, z)	Point in 2D/3D
Line(p1, p2)	Line through two points
Ray(p1, p2)	Ray from p1 through p2
Segment(p1, p2)	Line segment
Circle(center, radius)	Circle
Ellipse(center, hradius, vradius)	Ellipse
Triangle(p1, p2, p3)	Triangle
Polygon(*points)	N-sided polygon
RegularPolygon(center, radius, n)	Regular polygon
Curve(expr_tuple, param_range)	Parametric curve
Plane(p1, p2, p3)	3D plane

Methods: .area, .perimeter, .circumcircle, .incircle, .centroid, .distance(), .intersection(), .is_tangent(), .is_similar(), .is_congruent()

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Combinatorics -- sympy.combinatorics

Class	Description
Permutation([1, 0, 3, 2])	Permutation
Permutation.cyclic_form	Cycle notation
PermutationGroup([perms])	Permutation group
SymmetricGroup(n)	Symmetric group S_n
CyclicGroup(n)	Cyclic group Z_n
DihedralGroup(n)	Dihedral group D_n
AlternatingGroup(n)	Alternating group A_n
Partition([parts])	Integer partition
IntegerPartition(n)	All partitions of n
Subset(subset, superset)	Subset enumeration
GrayCode(n)	Gray code generation

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Statistics -- sympy.stats

Function / Class	Description
Normal('X', mu, sigma)	Normal random variable
Uniform('X', a, b)	Uniform random variable
Exponential('X', rate)	Exponential
Poisson('X', mu)	Poisson
Bernoulli('X', p)	Bernoulli
Binomial('X', n, p)	Binomial
Beta('X', a, b)	Beta
Gamma('X', k, theta)	Gamma
P(condition)	Probability
E(expr)	Expected value
variance(X)	Variance
std(X)	Standard deviation
covariance(X, Y)	Covariance
density(X)	Probability density function
cdf(X)	Cumulative distribution function
moment(X, n)	N-th moment
median(X)	Median
sample(X)	Generate random sample

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Logic -- sympy.logic

Function / Class	Description
And(a, b) / a & b	Logical AND
Or(a, b) / `a	b`
Not(a) / ~a	Logical NOT
Implies(a, b)	Implication
Equivalent(a, b)	Biconditional
Xor(a, b)	Exclusive OR
satisfiable(expr)	Find satisfying assignment
simplify_logic(expr)	Simplify logical expression
SOPform(variables, minterms)	Sum of products
POSform(variables, minterms)	Product of sums
truth_table(expr, variables)	Generate truth table

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Sets -- sympy.sets

Class	Description
FiniteSet(1, 2, 3)	Finite set
Interval(a, b)	Closed interval [a, b]
Interval.open(a, b)	Open interval (a, b)
S.Reals	Set of real numbers
S.Integers	Set of integers
S.Naturals	Set of natural numbers
S.Naturals0	Natural numbers including 0
S.Complexes	Set of complex numbers
S.EmptySet	Empty set
S.UniversalSet	Universal set
Union(A, B)	Set union
Intersection(A, B)	Set intersection
Complement(A, B)	Set difference A \ B
SymmetricDifference(A, B)	Symmetric difference
ProductSet(A, B)	Cartesian product
ImageSet(f, S)	Image of set under function
ConditionSet(x, condition, S)	Conditional set

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Printing & LaTeX

Function	Description
latex(expr)	LaTeX string
pretty(expr)	Unicode pretty-print string
mathml(expr)	MathML output
str(expr)	Python string
srepr(expr)	Internal representation
ccode(expr)	C code generation
fcode(expr)	Fortran code generation
jscode(expr)	JavaScript code generation
python(expr)	Python code string

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Physics -- sympy.physics

Module	Description
sympy.physics.units	Physical units and dimensions
sympy.physics.mechanics	Classical mechanics (Lagrangian, Hamiltonian)
sympy.physics.quantum	Quantum mechanics (operators, states, brakets)
sympy.physics.optics	Optics
sympy.physics.vector	Vector algebra in 3D reference frames
sympy.physics.hydrogen	Hydrogen atom wavefunctions
sympy.physics.paulialgebra	Pauli matrices
sympy.physics.wigner	Wigner symbols

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Not Available

• sympy.plotting (no display backend on iOS -- use matplotlib instead)
• Interactive features (init_printing, pprint renders in terminal only)