.. _calculus:
==========
Calculus
==========
This section covers how to do basic calculus tasks such as derivatives,
integrals, limits, and series expansions in SymPy. If you are not familiar
with the math of any part of this section, you may safely skip it.
>>> from sympy import *
>>> x, y, z = symbols('x y z')
>>> init_printing(use_unicode=True)
.. _tutorial-derivatives:
Derivatives
===========
To take derivatives, use the ``diff`` function.
>>> diff(cos(x), x)
-sin(x)
>>> diff(exp(x**2), x)
⎛ 2⎞
⎝x ⎠
2⋅x⋅ℯ
``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 number
after the variable. 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)
3 2 ⎛ 3 3 3 2 2 2 ⎞ x⋅y⋅z
x ⋅y ⋅⎝x ⋅y ⋅z + 14⋅x ⋅y ⋅z + 52⋅x⋅y⋅z + 48⎠⋅ℯ
>>> diff(expr, x, y, 2, z, 4)
3 2 ⎛ 3 3 3 2 2 2 ⎞ x⋅y⋅z
x ⋅y ⋅⎝x ⋅y ⋅z + 14⋅x ⋅y ⋅z + 52⋅x⋅y⋅z + 48⎠⋅ℯ
>>> diff(expr, x, y, y, z, 4)
3 2 ⎛ 3 3 3 2 2 2 ⎞ x⋅y⋅z
x ⋅y ⋅⎝x ⋅y ⋅z + 14⋅x ⋅y ⋅z + 52⋅x⋅y⋅z + 48⎠⋅ℯ
``diff`` can also be called as a method. The two ways of calling ``diff`` are
exactly the same, and are provided only for convenience.
>>> expr.diff(x, y, y, z, 4)
3 2 ⎛ 3 3 3 2 2 2 ⎞ x⋅y⋅z
x ⋅y ⋅⎝x ⋅y ⋅z + 14⋅x ⋅y ⋅z + 52⋅x⋅y⋅z + 48⎠⋅ℯ
To create an unevaluated derivative, use the ``Derivative`` class. It has the
same syntax as ``diff``.
>>> deriv = Derivative(expr, x, y, y, z, 4)
>>> deriv
7
∂ ⎛ x⋅y⋅z⎞
──────────⎝ℯ ⎠
4 2
∂z ∂y ∂x
To evaluate an unevaluated derivative, use the ``doit`` method.
>>> deriv.doit()
3 2 ⎛ 3 3 3 2 2 2 ⎞ x⋅y⋅z
x ⋅y ⋅⎝x ⋅y ⋅z + 14⋅x ⋅y ⋅z + 52⋅x⋅y⋅z + 48⎠⋅ℯ
These unevaluated objects are useful for delaying the evaluation of the
derivative, or for printing purposes. They are also used when SymPy does not
know how to compute the derivative of an expression (for example, if it
contains an undefined function, which are described in the :ref:`Solving
Differential Equations ` section).
Derivatives of unspecified order can be created using tuple ``(x, n)`` where
``n`` is the order of the derivative with respect to ``x``.
>>> m, n, a, b = symbols('m n a b')
>>> expr = (a*x + b)**m
>>> expr.diff((x, n))
n
∂ ⎛ m⎞
───⎝(a⋅x + b) ⎠
n
∂x
Integrals
=========
To compute an integral, use the ``integrate`` function. There are two kinds
of integrals, definite and indefinite. To compute an indefinite integral,
that is, an antiderivative, or primitive, just pass the variable after the
expression.
>>> integrate(cos(x), x)
sin(x)
Note that SymPy does not include the constant of integration. If you want it,
you can add one yourself, or rephrase your problem as a differential equation
and use ``dsolve`` to solve it, which does add the constant (see :ref:`tutorial-dsolve`).
.. sidebar:: Quick Tip
`\infty` in SymPy is ``oo`` (that's the lowercase letter "oh" twice). This
is because ``oo`` looks like `\infty`, and is easy to type.
To compute a definite integral, pass the argument ``(integration_variable,
lower_limit, upper_limit)``. For example, to compute
.. math::
\int_0^\infty e^{-x}\,dx,
we would do
>>> integrate(exp(-x), (x, 0, oo))
1
As with indefinite integrals, you can pass multiple limit tuples to perform a
multiple integral. For example, to compute
.. math::
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{- x^{2} - y^{2}}\, dx\, dy,
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.
>>> expr = integrate(x**x, x)
>>> print(expr)
Integral(x**x, x)
>>> expr
⌠
⎮ x
⎮ x dx
⌡
As with ``Derivative``, you can create an unevaluated integral using
``Integral``. To later evaluate this integral, call ``doit``.
>>> expr = Integral(log(x)**2, x)
>>> expr
⌠
⎮ 2
⎮ log (x) dx
⌡
>>> expr.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 heuristic pattern matching
type algorithms, a partial implementation of the `Risch algorithm
`_, and an algorithm using
`Meijer G-functions `_ that is
useful for computing integrals in terms of special functions, especially
definite integrals. Here is a sampling of some of the power of ``integrate``.
>>> integ = 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)
>>> integ
⌠
⎮ ⎛ 4 2 x 2 x x⎞ x
⎮ ⎝x + x ⋅ℯ - x - 2⋅x⋅ℯ - 2⋅x - ℯ ⎠⋅ℯ
⎮ ──────────────────────────────────────── dx
⎮ 2 2 ⎛ x ⎞
⎮ (x - 1) ⋅(x + 1) ⋅⎝ℯ + 1⎠
⌡
>>> integ.doit()
x
⎛ x ⎞ ℯ
log⎝ℯ + 1⎠ + ──────
2
x - 1
>>> integ = Integral(sin(x**2), x)
>>> integ
⌠
⎮ ⎛ 2⎞
⎮ sin⎝x ⎠ dx
⌡
>>> integ.doit()
⎛√2⋅x⎞
3⋅√2⋅√π⋅S⎜────⎟⋅Γ(3/4)
⎝ √π ⎠
──────────────────────
8⋅Γ(7/4)
>>> integ = Integral(x**y*exp(-x), (x, 0, oo))
>>> integ
∞
⌠
⎮ y -x
⎮ x ⋅ℯ dx
⌡
0
>>> integ.doit()
⎧ Γ(y + 1) for re(y) > -1
⎪
⎪∞
⎪⌠
⎨⎮ y -x
⎪⎮ x ⋅ℯ dx otherwise
⎪⌡
⎪0
⎩
This last example returned a ``Piecewise`` expression because the integral
does not converge unless `\Re(y) > -1.`
Numeric Integration
===================
Numeric integration is a method employed in mathematical analysis to estimate
the definite integral of a function across a simplified range. SymPy not only
facilitates symbolic integration but also provides support for numeric integration.
It leverages the precision capabilities of the ``mpmath`` library to enhance the
accuracy of numeric integration calculations.
>>> from sympy import Integral, Symbol, sqrt
>>> x = Symbol('x')
>>> integral = Integral(sqrt(2)*x, (x, 0, 1))
>>> integral
1
⌠
⎮ √2⋅x dx
⌡
0
>>> integral.evalf()
0.707106781186548
To compute the integral with a specified precision:
>>> integral.evalf(50)
0.70710678118654752440084436210484903928483593768847
Numeric integration becomes a viable approach in situations where symbolic integration
is impractical or impossible. This method allows for the computation of integrals
through numerical techniques, even when dealing with infinite intervals or integrands:
>>> Integral(exp(-(x ** 2)), (x, -oo, oo)).evalf()
1.77245385090552
>>> Integral(1 / sqrt(x), (x, 0, 1)).evalf()
2.00000000000000
Limits
======
SymPy can compute symbolic limits with the ``limit`` function. The syntax to compute
.. math::
\lim_{x\to x_0} f(x)
is ``limit(f(x), x, x0)``.
>>> limit(sin(x)/x, x, 0)
1
``limit`` should be used instead of ``subs`` whenever the point of evaluation
is a singularity. Even though SymPy 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``.
>>> expr = Limit((cos(x) - 1)/x, x, 0)
>>> expr
⎛cos(x) - 1⎞
lim ⎜──────────⎟
x─→0⁺⎝ x ⎠
>>> expr.doit()
0
To evaluate a limit at one side only, pass ``'+'`` or ``'-'`` as a fourth
argument to ``limit``. For example, to compute
.. math::
\lim_{x\to 0^+}\frac{1}{x},
do
>>> limit(1/x, x, 0, '+')
∞
As opposed to
>>> limit(1/x, x, 0, '-')
-∞
Series Expansion
================
SymPy can compute asymptotic series expansions of functions around a point. To
compute the expansion of `f(x)` around the point `x = x_0` terms of order
`x^n`, use ``f(x).series(x, x0, n)``. ``x0`` and ``n`` can be omitted, in
which case the defaults ``x0=0`` and ``n=6`` will be used.
>>> expr = exp(sin(x))
>>> expr.series(x, 0, 4)
2
x ⎛ 4⎞
1 + x + ── + O⎝x ⎠
2
The `O\left(x^4\right)` term 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` where `x \rightarrow \infty`). It means that all
x terms with power greater than or equal to `x^4` are omitted. Order terms
can be created and manipulated outside of ``series``. They automatically
absorb higher order terms.
>>> x + x**3 + x**6 + O(x**4)
3 ⎛ 4⎞
x + x + O⎝x ⎠
>>> x*O(1)
O(x)
If you do not want the order term, use the ``removeO`` method.
>>> expr.series(x, 0, 4).removeO()
2
x
── + x + 1
2
The ``O`` notation supports arbitrary limit points (other than 0):
>>> exp(x - 6).series(x, x0=6)
2 3 4 5
(x - 6) (x - 6) (x - 6) (x - 6) ⎛ 6 ⎞
-5 + ──────── + ──────── + ──────── + ──────── + x + O⎝(x - 6) ; x → 6⎠
2 6 24 120
.. _calculus-finite-differences:
Finite differences
==================
So far we have looked at expressions with analytic derivatives
and primitive functions respectively. But what if we want to have an
expression to estimate a derivative of a curve for which we lack a
closed form representation, or for which we don't know the functional
values for yet. One approach would be to use a finite difference
approach.
The simplest way the differentiate using finite differences is to use
the ``differentiate_finite`` function:
>>> f, g = symbols('f g', cls=Function)
>>> differentiate_finite(f(x)*g(x))
-f(x - 1/2)⋅g(x - 1/2) + f(x + 1/2)⋅g(x + 1/2)
If you already have a ``Derivative`` instance, you can use the
``as_finite_difference`` method to generate approximations of the
derivative to arbitrary order:
>>> f = Function('f')
>>> dfdx = f(x).diff(x)
>>> dfdx.as_finite_difference()
-f(x - 1/2) + f(x + 1/2)
here the first order derivative was approximated around x using a
minimum number of points (2 for 1st order derivative) evaluated
equidistantly using a step-size of 1. We can use arbitrary steps
(possibly containing symbolic expressions):
>>> f = Function('f')
>>> d2fdx2 = f(x).diff(x, 2)
>>> h = Symbol('h')
>>> d2fdx2.as_finite_difference([-3*h,-h,2*h])
f(-3⋅h) f(-h) 2⋅f(2⋅h)
─────── - ───── + ────────
2 2 2
5⋅h 3⋅h 15⋅h
If you are just interested in evaluating the weights, you can do so
manually:
>>> finite_diff_weights(2, [-3, -1, 2], 0)[-1][-1]
[1/5, -1/3, 2/15]
note that we only need the last element in the last sublist
returned from ``finite_diff_weights``. The reason for this is that
the function also generates weights for lower derivatives and
using fewer points (see the documentation of ``finite_diff_weights``
for more details).
If using ``finite_diff_weights`` directly looks complicated, and the
``as_finite_difference`` method of ``Derivative`` instances
is not flexible enough, you can use ``apply_finite_diff`` which
takes ``order``, ``x_list``, ``y_list`` and ``x0`` as parameters:
>>> x_list = [-3, 1, 2]
>>> y_list = symbols('a b c')
>>> apply_finite_diff(1, x_list, y_list, 0)
3⋅a b 2⋅c
- ─── - ─ + ───
20 4 5