Calculus
This section covers how to do basic calculus tasks such as derivatives, integrals, limits, and series expansions in Diofant.
Derivatives
To take derivatives, use the diff()
function.
>>> diff(cos(x))
-sin(x)
>>> diff(exp(x**2), x)
⎛ 2⎞
⎝x ⎠
2⋅ℯ ⋅x
diff()
can take multiple derivatives at
once. To take multiple derivatives, pass the variable as many times
as you wish to differentiate, or pass a tuple (variable and order).
For example, both of the following find the third derivative of \(x^4\).
>>> diff(x**4, x, x, x)
24⋅x
>>> diff(x**4, (x, 3))
24⋅x
You can also take derivatives with respect to many variables at once. Just pass each derivative in order, using the same syntax as for single variable derivatives. For example, each of the following will compute \(\frac{\partial^7}{\partial x\partial y^2\partial z^4} e^{x y z}\).
>>> expr = exp(x*y*z)
>>> diff(expr, x, y, y, z, z, z, z)
x⋅y⋅z 3 2 ⎛ 3 3 3 2 2 2 ⎞
ℯ ⋅x ⋅y ⋅⎝x ⋅y ⋅z + 14⋅x ⋅y ⋅z + 52⋅x⋅y⋅z + 48⎠
>>> diff(expr, x, (y, 2), (z, 4))
x⋅y⋅z 3 2 ⎛ 3 3 3 2 2 2 ⎞
ℯ ⋅x ⋅y ⋅⎝x ⋅y ⋅z + 14⋅x ⋅y ⋅z + 52⋅x⋅y⋅z + 48⎠
>>> diff(expr, x, y, y, (z, 4))
x⋅y⋅z 3 2 ⎛ 3 3 3 2 2 2 ⎞
ℯ ⋅x ⋅y ⋅⎝x ⋅y ⋅z + 14⋅x ⋅y ⋅z + 52⋅x⋅y⋅z + 48⎠
diff()
can also be called as a method
diff()
. The two ways of calling
diff()
are exactly the same, and are
provided only for convenience.
>>> expr.diff(x, y, y, (z, 4))
x⋅y⋅z 3 2 ⎛ 3 3 3 2 2 2 ⎞
ℯ ⋅x ⋅y ⋅⎝x ⋅y ⋅z + 14⋅x ⋅y ⋅z + 52⋅x⋅y⋅z + 48⎠
To create an unevaluated derivative, use the
Derivative
class. It has the same
syntax as diff()
.
>>> Derivative(expr, x, y, y, (z, 4))
7
∂ ⎛ x⋅y⋅z⎞
──────────⎝ℯ ⎠
4 2
∂z ∂y ∂x
Such unevaluated objects also used when Diofant does not know how to compute the derivative of an expression.
To evaluate an unevaluated derivative, use the
doit()
method.
>>> _.doit()
x⋅y⋅z 3 2 ⎛ 3 3 3 2 2 2 ⎞
ℯ ⋅x ⋅y ⋅⎝x ⋅y ⋅z + 14⋅x ⋅y ⋅z + 52⋅x⋅y⋅z + 48⎠
Integrals
To compute an integral, use the integrate()
function. There are two kinds of integrals, definite and indefinite. To
compute an indefinite integral, do
>>> integrate(cos(x))
sin(x)
Note
For indefinite integrals, Diofant does not include the constant of integration.
For example, to compute a definite integral
we would do
>>> integrate(exp(-x), (x, 0, oo))
1
Tip
\(\infty\) in Diofant is oo
(that’s the lowercase letter “oh” twice).
As with indefinite integrals, you can pass multiple limit tuples to perform a multiple integral. For example, to compute
do
>>> integrate(exp(-x**2 - y**2), (x, -oo, oo), (y, -oo, oo))
π
If integrate()
is unable to compute an
integral, it returns an unevaluated
Integral
object.
>>> integrate(x**x)
⌠
⎮ x
⎮ x dx
⌡
>>> print(_)
Integral(x**x, x)
As with Derivative
, you can create an
unevaluated integral directly. To later evaluate this integral, call
doit()
.
>>> Integral(log(x)**2)
⌠
⎮ 2
⎮ log (x) dx
⌡
>>> _.doit()
2
x⋅log (x) - 2⋅x⋅log(x) + 2⋅x
integrate()
uses powerful algorithms that
are always improving to compute both definite and indefinite integrals,
including a partial implementation of the Risch algorithm
>>> Integral((x**4 + x**2*exp(x) - x**2 - 2*x*exp(x) - 2*x -
... exp(x))*exp(x)/((x - 1)**2*(x + 1)**2*(exp(x) + 1)))
⌠
⎮ x ⎛ x 2 x x 4 2 ⎞
⎮ ℯ ⋅⎝ℯ ⋅x - 2⋅ℯ ⋅x - ℯ + x - x - 2⋅x⎠
⎮ ──────────────────────────────────────── dx
⎮ ⎛ x ⎞ 2 2
⎮ ⎝ℯ + 1⎠⋅(x - 1) ⋅(x + 1)
⌡
>>> _.doit()
x
ℯ ⎛ x ⎞
────── + log⎝ℯ + 1⎠
2
x - 1
and an algorithm using Meijer G-functions that is useful for computing integrals in terms of special functions, especially definite integrals
>>> Integral(sin(x**2))
⌠
⎮ ⎛ 2⎞
⎮ sin⎝x ⎠ dx
⌡
>>> _.doit()
⎛ ___ ⎞
___ ___ ⎜╲╱ 2 ⋅x⎟
3⋅╲╱ 2 ⋅╲╱ π ⋅fresnels⎜───────⎟⋅Γ(3/4)
⎜ ___ ⎟
⎝ ╲╱ π ⎠
──────────────────────────────────────
8⋅Γ(7/4)
>>> Integral(x**y*exp(-x), (x, 0, oo))
∞
⌠
⎮ -x y
⎮ ℯ ⋅x dx
⌡
0
>>> _.doit()
⎧ Γ(y + 1) for -re(y) < 1
⎪
⎪∞
⎪⌠
⎨⎮ -x y
⎪⎮ ℯ ⋅x dx otherwise
⎪⌡
⎪0
⎩
This last example returned a
Piecewise
expression because
the integral does not converge unless \(\Re(y) > 1.\)
Sums and Products
Much like integrals, there are
summation()
and
product()
to compute sums and
products respectively.
>>> summation(2**x, (x, 0, y - 1))
y
2 - 1
>>> product(z, (x, 1, y))
y
z
Unevaluated sums and products are represented by
Sum
and
Product
classes.
>>> Sum(1, (x, 1, z))
z
___
╲
╲ 1
╱
╱
‾‾‾
x = 1
>>> _.doit()
z
Limits
Diofant can compute symbolic limits with the
limit()
function. To compute a directional limit
do
>>> limit(sin(x)/x, x, 0)
1
limit()
should be used instead of
subs()
whenever the point of evaluation is a
singularity. Even though Diofant has objects to represent \(\infty\), using them
for evaluation is not reliable because they do not keep track of things like
rate of growth. Also, things like \(\infty - \infty\) and
\(\frac{\infty}{\infty}\) return \(\mathrm{nan}\) (not-a-number). For example
>>> expr = x**2/exp(x)
>>> expr.subs({x: oo})
nan
>>> limit(expr, x, oo)
0
Like Derivative
and
Integral
,
limit()
has an unevaluated counterpart,
Limit
. To evaluate it, use
doit()
.
>>> Limit((cos(x) - 1)/x, x, 0)
cos(x) - 1
lim ──────────
x─→0⁺ x
>>> _.doit()
0
To change side, pass '-'
as a third argument to
limit()
. For example, to compute
do
>>> limit(1/x, x, 0, dir=1)
-∞
You can also evaluate bidirectional limit
>>> limit(sin(x)/x, x, 0, dir=Reals)
1
>>> limit(1/x, x, 0, dir=Reals)
Traceback (most recent call last):
...
PoleError: left and right limits for expression 1/x at point x=0 seems to be not equal
Series Expansion
Diofant can compute asymptotic series expansions of functions around a point.
>>> exp(sin(x)).series(x, 0, 4)
2
x ⎛ 4⎞
1 + x + ── + O⎝x ⎠
2
The \(O\left (x^4\right )\) term, an instance of O
at the end represents the Landau order term at \(x=0\) (not to be confused with
big O notation used in computer science, which generally represents the Landau
order term at \(x=\infty\)). Order terms can be created and manipulated outside
of series()
.
>>> x + x**3 + x**6 + O(x**4)
3 ⎛ 4⎞
x + x + O⎝x ⎠
>>> x*O(1, x)
O(x)
If you do not want the order term, use the
removeO()
method.
>>> exp(x).series(x, 0, 3).removeO()
2
x
── + x + 1
2
The O
notation supports arbitrary limit points:
>>> exp(x - 1).series(x, x0=1)
2 3 4 5
(x - 1) (x - 1) (x - 1) (x - 1) ⎛ 6 ⎞
──────── + ──────── + ──────── + ──────── + x + O⎝(x - 1) ; x → 1⎠
2 6 24 120