def limit(f, dir=None, taylor=False, **argv):
r"""
Return the limit as the variable `v` approaches `a`
from the given direction.
::
limit(expr, x = a)
limit(expr, x = a, dir='above')
INPUT:
- ``dir`` - (default: None); dir may have the value
'plus' (or 'above') for a limit from above, 'minus' (or 'below')
for a limit from below, or may be omitted (implying a two-sided
limit is to be computed).
- ``taylor`` - (default: False); if True, use Taylor
series, which allows more limits to be computed (but may also
crash in some obscure cases due to bugs in Maxima).
- ``\*\*argv`` - 1 named parameter
ALIAS: You can also use lim instead of limit.
EXAMPLES::
sage: limit(sin(x)/x, x=0)
1
sage: limit(exp(x), x=oo)
+Infinity
sage: lim(exp(x), x=-oo)
0
sage: lim(1/x, x=0)
Infinity
sage: limit(sqrt(x^2+x+1)+x, taylor=True, x=-oo)
-1/2
sage: limit((tan(sin(x)) - sin(tan(x)))/x^7, taylor=True, x=0)
1/30
Sage does not know how to do this limit (which is 0), so it returns
it unevaluated::
sage: lim(exp(x^2)*(1-erf(x)), x=infinity)
limit(-e^(x^2)*erf(x) + e^(x^2), x, +Infinity)
"""
if not isinstance(f, Expression):
f = SR(f)
return f.limit(dir=dir, taylor=taylor, **argv)
def limit(f, dir=None, taylor=False, **argv):
r"""
Return the limit as the variable `v` approaches `a`
from the given direction.
::
limit(expr, x = a)
limit(expr, x = a, dir='above')
INPUT:
- ``dir`` - (default: None); dir may have the value
'plus' (or 'above') for a limit from above, 'minus' (or 'below')
for a limit from below, or may be omitted (implying a two-sided
limit is to be computed).
- ``taylor`` - (default: False); if True, use Taylor
series, which allows more limits to be computed (but may also
crash in some obscure cases due to bugs in Maxima).
- ``\*\*argv`` - 1 named parameter
ALIAS: You can also use lim instead of limit.
EXAMPLES::
sage: limit(sin(x)/x, x=0)
1
sage: limit(exp(x), x=oo)
+Infinity
sage: lim(exp(x), x=-oo)
0
sage: lim(1/x, x=0)
Infinity
sage: limit(sqrt(x^2+x+1)+x, taylor=True, x=-oo)
-1/2
sage: limit((tan(sin(x)) - sin(tan(x)))/x^7, taylor=True, x=0)
1/30
Sage does not know how to do this limit (which is 0), so it returns
it unevaluated::
sage: lim(exp(x^2)*(1-erf(x)), x=infinity)
-limit((erf(x) - 1)*e^(x^2), x, +Infinity)
"""
if not isinstance(f, Expression):
f = SR(f)
return f.limit(dir=dir, taylor=taylor, **argv)
def taylor(f, *args):
"""
Expands self in a truncated Taylor or Laurent series in the
variable `v` around the point `a`, containing terms
through `(x - a)^n`. Functions in more variables are also
supported.
INPUT:
- ``*args`` - the following notation is supported
- ``x, a, n`` - variable, point, degree
- ``(x, a), (y, b), ..., n`` - variables with points, degree of polynomial
EXAMPLES::
sage: var('x,k,n')
(x, k, n)
sage: taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6)
-1/720*(45*k^6 - 60*k^4 + 16*k^2)*x^6 - 1/24*(3*k^4 - 4*k^2)*x^4 - 1/2*k^2*x^2 + 1
::
sage: taylor ((x + 1)^n, x, 0, 4)
1/24*(n^4 - 6*n^3 + 11*n^2 - 6*n)*x^4 + 1/6*(n^3 - 3*n^2 + 2*n)*x^3 + 1/2*(n^2 - n)*x^2 + n*x + 1
::
sage: taylor ((x + 1)^n, x, 0, 4)
1/24*(n^4 - 6*n^3 + 11*n^2 - 6*n)*x^4 + 1/6*(n^3 - 3*n^2 + 2*n)*x^3 + 1/2*(n^2 - n)*x^2 + n*x + 1
Taylor polynomial in two variables::
sage: x,y=var('x y'); taylor(x*y^3,(x,1),(y,-1),4)
(y + 1)^3*(x - 1) + (y + 1)^3 - 3*(y + 1)^2*(x - 1) - 3*(y + 1)^2 + 3*(y + 1)*(x - 1) - x + 3*y + 3
"""
if not isinstance(f, Expression):
f = SR(f)
return f.taylor(*args)
def taylor(f, *args):
"""
Expands self in a truncated Taylor or Laurent series in the
variable `v` around the point `a`, containing terms
through `(x - a)^n`. Functions in more variables are also
supported.
INPUT:
- ``*args`` - the following notation is supported
- ``x, a, n`` - variable, point, degree
- ``(x, a), (y, b), ..., n`` - variables with points, degree of polynomial
EXAMPLES::
sage: var('x,k,n')
(x, k, n)
sage: taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6)
-1/720*(45*k^6 - 60*k^4 + 16*k^2)*x^6 - 1/24*(3*k^4 - 4*k^2)*x^4 - 1/2*k^2*x^2 + 1
::
sage: taylor ((x + 1)^n, x, 0, 4)
1/24*(n^4 - 6*n^3 + 11*n^2 - 6*n)*x^4 + 1/6*(n^3 - 3*n^2 + 2*n)*x^3 + 1/2*(n^2 - n)*x^2 + n*x + 1
::
sage: taylor ((x + 1)^n, x, 0, 4)
1/24*(n^4 - 6*n^3 + 11*n^2 - 6*n)*x^4 + 1/6*(n^3 - 3*n^2 + 2*n)*x^3 + 1/2*(n^2 - n)*x^2 + n*x + 1
Taylor polynomial in two variables::
sage: x,y=var('x y'); taylor(x*y^3,(x,1),(y,-1),4)
(x - 1)*(y + 1)^3 - 3*(x - 1)*(y + 1)^2 + (y + 1)^3 + 3*(x - 1)*(y + 1) - 3*(y + 1)^2 - x + 3*y + 3
"""
if not isinstance(f, Expression):
f = SR(f)
return f.taylor(*args)
def derivative(f, *args, **kwds):
"""
The derivative of `f`.
Repeated differentiation is supported by the syntax given in the
examples below.
ALIAS: diff
EXAMPLES: We differentiate a callable symbolic function::
sage: f(x,y) = x*y + sin(x^2) + e^(-x)
sage: f
(x, y) |--> x*y + e^(-x) + sin(x^2)
sage: derivative(f, x)
(x, y) |--> 2*x*cos(x^2) + y - e^(-x)
sage: derivative(f, y)
(x, y) |--> x
We differentiate a polynomial::
sage: t = polygen(QQ, 't')
sage: f = (1-t)^5; f
-t^5 + 5*t^4 - 10*t^3 + 10*t^2 - 5*t + 1
sage: derivative(f)
-5*t^4 + 20*t^3 - 30*t^2 + 20*t - 5
sage: derivative(f, t)
-5*t^4 + 20*t^3 - 30*t^2 + 20*t - 5
sage: derivative(f, t, t)
-20*t^3 + 60*t^2 - 60*t + 20
sage: derivative(f, t, 2)
-20*t^3 + 60*t^2 - 60*t + 20
sage: derivative(f, 2)
-20*t^3 + 60*t^2 - 60*t + 20
We differentiate a symbolic expression::
sage: var('a x')
(a, x)
sage: f = exp(sin(a - x^2))/x
sage: derivative(f, x)
-2*e^(sin(-x^2 + a))*cos(-x^2 + a) - e^(sin(-x^2 + a))/x^2
sage: derivative(f, a)
e^(sin(-x^2 + a))*cos(-x^2 + a)/x
Syntax for repeated differentiation::
sage: R.<u, v> = PolynomialRing(QQ)
sage: f = u^4*v^5
sage: derivative(f, u)
4*u^3*v^5
sage: f.derivative(u) # can always use method notation too
4*u^3*v^5
::
sage: derivative(f, u, u)
12*u^2*v^5
sage: derivative(f, u, u, u)
24*u*v^5
sage: derivative(f, u, 3)
24*u*v^5
::
sage: derivative(f, u, v)
20*u^3*v^4
sage: derivative(f, u, 2, v)
60*u^2*v^4
sage: derivative(f, u, v, 2)
80*u^3*v^3
sage: derivative(f, [u, v, v])
80*u^3*v^3
"""
try:
return f.derivative(*args, **kwds)
except AttributeError:
pass
if not isinstance(f, Expression):
f = SR(f)
return f.derivative(*args, **kwds)
def integral(f, *args, **kwds):
r"""
The integral of `f`.
EXAMPLES::
sage: integral(sin(x), x)
-cos(x)
sage: integral(sin(x)^2, x, pi, 123*pi/2)
121/4*pi
sage: integral( sin(x), x, 0, pi)
2
We integrate a symbolic function::
sage: f(x,y,z) = x*y/z + sin(z)
sage: integral(f, z)
(x, y, z) |--> x*y*log(z) - cos(z)
::
sage: var('a,b')
(a, b)
sage: assume(b-a>0)
sage: integral( sin(x), x, a, b)
cos(a) - cos(b)
sage: forget()
::
sage: integral(x/(x^3-1), x)
1/3*sqrt(3)*arctan(1/3*(2*x + 1)*sqrt(3)) + 1/3*log(x - 1) - 1/6*log(x^2 + x + 1)
::
sage: integral( exp(-x^2), x )
1/2*sqrt(pi)*erf(x)
We define the Gaussian, plot and integrate it numerically and
symbolically::
sage: f(x) = 1/(sqrt(2*pi)) * e^(-x^2/2)
sage: P = plot(f, -4, 4, hue=0.8, thickness=2)
sage: P.show(ymin=0, ymax=0.4)
sage: numerical_integral(f, -4, 4) # random output
(0.99993665751633376, 1.1101527003413533e-14)
sage: integrate(f, x)
x |--> 1/2*erf(1/2*sqrt(2)*x)
You can have Sage calculate multiple integrals. For example,
consider the function `exp(y^2)` on the region between the
lines `x=y`, `x=1`, and `y=0`. We find the
value of the integral on this region using the command::
sage: area = integral(integral(exp(y^2),x,0,y),y,0,1); area
1/2*e - 1/2
sage: float(area)
0.859140914229522...
We compute the line integral of `\sin(x)` along the arc of
the curve `x=y^4` from `(1,-1)` to
`(1,1)`::
sage: t = var('t')
sage: (x,y) = (t^4,t)
sage: (dx,dy) = (diff(x,t), diff(y,t))
sage: integral(sin(x)*dx, t,-1, 1)
0
sage: restore('x,y') # restore the symbolic variables x and y
Sage is unable to do anything with the following integral::
sage: integral( exp(-x^2)*log(x), x )
integrate(e^(-x^2)*log(x), x)
Note, however, that::
sage: integral( exp(-x^2)*ln(x), x, 0, oo)
-1/4*(euler_gamma + 2*log(2))*sqrt(pi)
This definite integral is easy::
sage: integral( ln(x)/x, x, 1, 2)
1/2*log(2)^2
Sage can't do this elliptic integral (yet)::
sage: integral(1/sqrt(2*t^4 - 3*t^2 - 2), t, 2, 3)
integrate(1/sqrt(2*t^4 - 3*t^2 - 2), t, 2, 3)
A double integral::
sage: y = var('y')
sage: integral(integral(x*y^2, x, 0, y), y, -2, 2)
32/5
This illustrates using assumptions::
sage: integral(abs(x), x, 0, 5)
25/2
sage: a = var("a")
sage: integral(abs(x), x, 0, a)
1/2*a*abs(a)
sage: integral(abs(x)*x, x, 0, a)
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before integral evaluation
*may* help (example of legal syntax is 'assume(a>0)',
see `assume?` for more details)
Is a positive, negative, or zero?
sage: assume(a>0)
sage: integral(abs(x)*x, x, 0, a)
1/3*a^3
sage: forget() # forget the assumptions.
We integrate and differentiate a huge mess::
sage: f = (x^2-1+3*(1+x^2)^(1/3))/(1+x^2)^(2/3)*x/(x^2+2)^2
sage: g = integral(f, x)
sage: h = f - diff(g, x)
::
sage: [float(h(i)) for i in range(5)] #random
[0.0,
-1.1102230246251565e-16,
-5.5511151231257827e-17,
-5.5511151231257827e-17,
-6.9388939039072284e-17]
sage: h.factor()
0
sage: bool(h == 0)
True
"""
try:
return f.integral(*args, **kwds)
except AttributeError:
pass
if not isinstance(f, Expression):
f = SR(f)
return f.integral(*args, **kwds)
def derivative(f, *args, **kwds):
"""
The derivative of `f`.
Repeated differentiation is supported by the syntax given in the
examples below.
ALIAS: diff
EXAMPLES: We differentiate a callable symbolic function::
sage: f(x,y) = x*y + sin(x^2) + e^(-x)
sage: f
(x, y) |--> x*y + e^(-x) + sin(x^2)
sage: derivative(f, x)
(x, y) |--> 2*x*cos(x^2) + y - e^(-x)
sage: derivative(f, y)
(x, y) |--> x
We differentiate a polynomial::
sage: t = polygen(QQ, 't')
sage: f = (1-t)^5; f
-t^5 + 5*t^4 - 10*t^3 + 10*t^2 - 5*t + 1
sage: derivative(f)
-5*t^4 + 20*t^3 - 30*t^2 + 20*t - 5
sage: derivative(f, t)
-5*t^4 + 20*t^3 - 30*t^2 + 20*t - 5
sage: derivative(f, t, t)
-20*t^3 + 60*t^2 - 60*t + 20
sage: derivative(f, t, 2)
-20*t^3 + 60*t^2 - 60*t + 20
sage: derivative(f, 2)
-20*t^3 + 60*t^2 - 60*t + 20
We differentiate a symbolic expression::
sage: var('a x')
(a, x)
sage: f = exp(sin(a - x^2))/x
sage: derivative(f, x)
-2*cos(-x^2 + a)*e^(sin(-x^2 + a)) - e^(sin(-x^2 + a))/x^2
sage: derivative(f, a)
cos(-x^2 + a)*e^(sin(-x^2 + a))/x
Syntax for repeated differentiation::
sage: R.<u, v> = PolynomialRing(QQ)
sage: f = u^4*v^5
sage: derivative(f, u)
4*u^3*v^5
sage: f.derivative(u) # can always use method notation too
4*u^3*v^5
::
sage: derivative(f, u, u)
12*u^2*v^5
sage: derivative(f, u, u, u)
24*u*v^5
sage: derivative(f, u, 3)
24*u*v^5
::
sage: derivative(f, u, v)
20*u^3*v^4
sage: derivative(f, u, 2, v)
60*u^2*v^4
sage: derivative(f, u, v, 2)
80*u^3*v^3
sage: derivative(f, [u, v, v])
80*u^3*v^3
"""
try:
return f.derivative(*args, **kwds)
except AttributeError:
pass
if not isinstance(f, Expression):
f = SR(f)
return f.derivative(*args, **kwds)
def integral(f, *args, **kwds):
r"""
The integral of `f`.
EXAMPLES::
sage: integral(sin(x), x)
-cos(x)
sage: integral(sin(x)^2, x, pi, 123*pi/2)
121/4*pi
sage: integral( sin(x), x, 0, pi)
2
We integrate a symbolic function::
sage: f(x,y,z) = x*y/z + sin(z)
sage: integral(f, z)
(x, y, z) |--> x*y*log(z) - cos(z)
::
sage: var('a,b')
(a, b)
sage: assume(b-a>0)
sage: integral( sin(x), x, a, b)
cos(a) - cos(b)
sage: forget()
::
sage: integral(x/(x^3-1), x)
1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/6*log(x^2 + x + 1) + 1/3*log(x - 1)
::
sage: integral( exp(-x^2), x )
1/2*sqrt(pi)*erf(x)
We define the Gaussian, plot and integrate it numerically and
symbolically::
sage: f(x) = 1/(sqrt(2*pi)) * e^(-x^2/2)
sage: P = plot(f, -4, 4, hue=0.8, thickness=2)
sage: P.show(ymin=0, ymax=0.4)
sage: numerical_integral(f, -4, 4) # random output
(0.99993665751633376, 1.1101527003413533e-14)
sage: integrate(f, x)
x |--> 1/2*erf(1/2*sqrt(2)*x)
You can have Sage calculate multiple integrals. For example,
consider the function `exp(y^2)` on the region between the
lines `x=y`, `x=1`, and `y=0`. We find the
value of the integral on this region using the command::
sage: area = integral(integral(exp(y^2),x,0,y),y,0,1); area
1/2*e - 1/2
sage: float(area)
0.859140914229522...
We compute the line integral of `\sin(x)` along the arc of
the curve `x=y^4` from `(1,-1)` to
`(1,1)`::
sage: t = var('t')
sage: (x,y) = (t^4,t)
sage: (dx,dy) = (diff(x,t), diff(y,t))
sage: integral(sin(x)*dx, t,-1, 1)
0
sage: restore('x,y') # restore the symbolic variables x and y
Sage is unable to do anything with the following integral::
sage: integral( exp(-x^2)*log(x), x )
integrate(e^(-x^2)*log(x), x)
Note, however, that::
sage: integral( exp(-x^2)*ln(x), x, 0, oo)
-1/4*sqrt(pi)*(euler_gamma + 2*log(2))
This definite integral is easy::
sage: integral( ln(x)/x, x, 1, 2)
1/2*log(2)^2
Sage can't do this elliptic integral (yet)::
sage: integral(1/sqrt(2*t^4 - 3*t^2 - 2), t, 2, 3)
integrate(1/sqrt(2*t^4 - 3*t^2 - 2), t, 2, 3)
A double integral::
sage: y = var('y')
sage: integral(integral(x*y^2, x, 0, y), y, -2, 2)
32/5
This illustrates using assumptions::
sage: integral(abs(x), x, 0, 5)
25/2
sage: a = var("a")
sage: integral(abs(x), x, 0, a)
1/2*a*abs(a)
sage: integral(abs(x)*x, x, 0, a)
Traceback (most recent call last):
...
ValueError: Computation failed since Maxima requested additional
constraints; using the 'assume' command before integral evaluation
*may* help (example of legal syntax is 'assume(a>0)',
see `assume?` for more details)
Is a positive, negative or zero?
sage: assume(a>0)
sage: integral(abs(x)*x, x, 0, a)
1/3*a^3
sage: forget() # forget the assumptions.
We integrate and differentiate a huge mess::
sage: f = (x^2-1+3*(1+x^2)^(1/3))/(1+x^2)^(2/3)*x/(x^2+2)^2
sage: g = integral(f, x)
sage: h = f - diff(g, x)
::
sage: [float(h(i)) for i in range(5)] #random
[0.0,
-1.1102230246251565e-16,
-5.5511151231257827e-17,
-5.5511151231257827e-17,
-6.9388939039072284e-17]
sage: h.factor()
0
sage: bool(h == 0)
True
"""
try:
return f.integral(*args, **kwds)
except AttributeError:
pass
if not isinstance(f, Expression):
f = SR(f)
return f.integral(*args, **kwds)