Sage Notebook, Partial Fraction Decomposition

Sage Notebook is a free web-based, open-source, mathematical worksheet development environment. It is an alternative to Magma, Maple, Mathematica, and MATLAB. It can perform symbolic algebraic and calculus operations.

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Here is an example of an algebraic computation in Sage….

Partial fraction decomposition with variable coefficients:

f(a,b,x)=1/((x+a)*(x+b))
f.partial_fraction(x)

Evaluate

(a, b, x) |–> 1/((b + x)*(a – b)) – 1/((a + x)*(a – b))

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Here is an example for an inverse Laplace transform:

var(‘a b s t’)
assume(b>0)
inverse_laplace((s+a)/((s+a)^2+b^2),s,t)

Evaluate

e^(-a*t)*cos(b*t)

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Here is an example for the solution of a second-order ODE for the free vibration response of a single-degree-of-freedom system.

damp,omegan,t = var(‘damp omegan t’)
x=function(‘x’,t)
assume(damp-1 0)
desolve(diff(x,t,2)+2*damp*omegan*diff(x,t,1)+omegan^2*x==0,x,ivar=t,contrib_ode=True)

Evaluate

(k1*sin(1/2*sqrt(-4*damp^2*omegan^2 + 4*omegan^2)*t) +
k2*cos(1/2*sqrt(-4*damp^2*omegan^2 + 4*omegan^2)*t))*e^(-damp*omegan*t)

Note that:

k1 and k2 are coefficients which must be determined from the initial conditions.

omegan is the undamped natural frequency (radians/time).

1/2*sqrt(-4*damp^2*omegan^2 + 4*omegan^2) is the damped natural frequency.

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Indefinite intergal example:

x, a, b = var(‘x a b’)
integral(a*(sin(b*x))^2, x)

Evaluate

1/4*(2*b*x – sin(2*b*x))*a/b

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Definite intergal with variable coefficients example:

x,a,b = var(‘x,a,b’)
integral(a*(sin(b*x))^2, x, 0, pi)

Evaluate

1/4*(2*pi*b – sin(2*pi*b))*a/b

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Tom Irvine