SymPy Gamma

Fractions

• Simplify fractions 242/33
• Rationalize repeating decimals 0.[123]

Approximations

• pi
• E
• exp(pi)

• x
• (x+2)/((x+3)(x-4))
• simplify((x**2 - 4)/((x+3)(x-2)))

Polynomial and Rational Functions

• Polynomial division div(x**2 - 4 + x, x-2)
• Greatest common divisor gcd(2*x**2 + 6*x, 12*x)
• …and least common multiple lcm(2*x**2 + 6*x, 12*x)
• Factorization factor(x**4/2 + 5*x**3/12 - x**2/3)
• Multivariate factorization factor(x**2 + 4*x*y + 4*y**2)
• Symbolic roots solve(x**2 + 4*x*y + 4*y**2)
• solve(x**2 + 4*x*y + 4*y**2, y)
• Complex roots solve(x**2 + 4*x + 181, x)
• Irrational roots solve(x**3 + 4*x + 181, x)
• Systems of equations solve_poly_system([y**2 - x**3 + 1, y*x], x, y)

• sin(2x)
• tan(1 + x)

Limits

• limit(tan(x), x, pi/2)
• limit(tan(x), x, pi/2, dir="-")

Derivatives

• Derive the product rule diff(f(x)*g(x)*h(x))
• …as well as the quotient rule diff(f(x)/g(x))
• Get steps for derivatives diff((sin(x) * x^2) / (1 + tan(cot(x))))
• Multiple ways to derive functions diff(cot(xy), y)
• Implicit derivatives, too diff(y(x)^2 - 5sin(x), x)

Integrals

• integrate(tan(x))
• Multiple variables integrate(2*x + y, y)
• Limits of integration integrate(2*x + y, (x, 1, 3))
• integrate(2*x + y, (x, 1, 3), (y, 2, 4))
• Improper integrals integrate(tan(x), (x, 0, pi/2))
• Exact answers integrate(1/(x**2 + 1), (x, 0, oo))
• Get steps for integrals integrate(exp(x) / (1 + exp(2x)))
• integrate(1 /((x+1)(x+3)(x+5)))
• integrate((2x+3)**7)

Series

• series(sin(x), x, pi/2)

• 1006!
• factorint(12321)
• Calculate the 42^nd prime prime(42)
• Calculate \( \varphi(x) \), the Euler totient function totient(42)
• isprime(12321)
• First prime greater than 42 nextprime(42)

Diophantine Equations

• diophantine(x**2 - 4*x*y + 8*y**2 - 3*x + 7*y - 5)
• diophantine(2*x + 3*y - 5)
• diophantine(3*x**2 + 4*y**2 - 5*z**2 + 4*x*y - 7*y*z + 7*z*x)

Boolean Logic

• (x | y) & (x | ~y) & (~x | y)
• x & ~x

Recurrences

• Solve a recurrence relation rsolve(y(n+2)-y(n+1)-y(n), y(n))
• Specify initial conditions rsolve(y(n+2)-y(n+1)-y(n), y(n), {y(0): 0, y(1): 1})

Summation

• Sum(k,(k,1,m))
• Sum(x**k,(k,0,oo))
• Product(k**2,(k,1,m))
• summation(1/2**i, (i, 0, oo))
• product(i, (i, 1, k), (k, 1, n))

• plot(sin(x) + cos(2x))
• Multiple plots plot([x, x^2, x^3, x^4])
• Polar plots plot(r=1-sin(theta))
• Parametric plots plot(x=cos(t), y=sin(t))
• Multiple plot types plot(y=x,y1=x^2,r=cos(theta),r1=sin(theta))

• Documentation for functions factorial2
• sympify
• bernoulli