Sage implements several generalized functions (also known as distributions) such as Dirac delta, Heaviside step functions. These generalized functions can be manipulated within Sage like any other symbolic functions.
REFERENCES:
AUTHORS:
EXAMPLES:
Dirac delta function:
sage: dirac_delta(x)
dirac_delta(x)
Heaviside step function:
sage: heaviside(x)
heaviside(x)
Unit step function:
sage: unit_step(x)
unit_step(x)
The Dirac delta (generalized) function, (dirac_delta(x)).
INPUT:
DEFINITION:
Dirac delta function , is defined in Sage as:
for real
and
Its alternate definition with respect to an arbitrary test
function is
EXAMPLES:
sage: dirac_delta(1)
0
sage: dirac_delta(0)
dirac_delta(0)
sage: dirac_delta(x)
dirac_delta(x)
The Dirac delta (generalized) function, dirac_delta(x).
INPUT:
EXAMPLES:
sage: dirac_delta(1)
0
sage: dirac_delta(0)
dirac_delta(0)
sage: dirac_delta(x)
dirac_delta(x)
INPUT:
EXAMPLES:
sage: dirac_delta(1)
0
sage: dirac_delta(0)
dirac_delta(0)
sage: dirac_delta(x)
dirac_delta(x)
sage: dirac_delta(exp(-10000000000000000000))
0
Evaluation test:
sage: dirac_delta(x).subs(x=1)
0
The Heaviside step function, (heaviside(x)).
INPUT:
DEFINITION:
The Heaviside step function, is defined in Sage as:
for
and
for
EXAMPLES:
sage: heaviside(-1)
0
sage: heaviside(1)
1
sage: heaviside(0)
heaviside(0)
sage: heaviside(x)
heaviside(x)
The Heaviside step function, heaviside(x).
INPUT:
EXAMPLES:
sage: heaviside(-1)
0
sage: heaviside(1)
1
sage: heaviside(0)
heaviside(0)
sage: heaviside(x)
heaviside(x)
Derivative of Heaviside step function
EXAMPLES:
sage: heaviside(x).diff(x)
dirac_delta(x)
INPUT:
EXAMPLES:
sage: heaviside(-1/2)
0
sage: heaviside(1)
1
sage: heaviside(0)
heaviside(0)
sage: heaviside(x)
heaviside(x)
sage: heaviside(exp(-1000000000000000000000))
1
Evaluation test:
sage: heaviside(x).subs(x=1)
1
sage: heaviside(x).subs(x=-1)
0
The unit step function, (unit_step(x)).
INPUT:
DEFINITION:
The unit step function, is defined in Sage as:
for
and
for
EXAMPLES:
sage: unit_step(-1)
0
sage: unit_step(1)
1
sage: unit_step(0)
1
sage: unit_step(x)
unit_step(x)
The unit step function, unit_step(x).
INPUT:
EXAMPLES:
sage: unit_step(-1) 0 sage: unit_step(1) 1 sage: unit_step(0) 1 sage: unit_step(x) unit_step(x)
Derivative of unit step function
EXAMPLES:
sage: unit_step(x).diff(x)
dirac_delta(x)
INPUT:
EXAMPLES:
sage: unit_step(-1)
0
sage: unit_step(1)
1
sage: unit_step(0)
1
sage: unit_step(x)
unit_step(x)
sage: unit_step(-exp(-10000000000000000000))
0
Evaluation test:
sage: unit_step(x).subs(x=1)
1
sage: unit_step(x).subs(x=0)
1