Elementary

This module implements elementary functions such as trigonometric, hyperbolic as well as functions like Abs, Max, sqrt etc.

diofant.functions.elementary.complexes

re

class diofant.functions.elementary.complexes.re(arg)[source]

Returns real part of expression.

This function performs only elementary analysis and so it will fail to decompose properly more complicated expressions. If completely simplified result is needed then use Basic.as_real_imag() or perform complex expansion on instance of this function.

Examples

>>> re(2*E)
2*E
>>> re(2*I + 17)
17
>>> re(2*I)
0
>>> re(im(x) + x*I + 2)
2

as_real_imag(deep=True, **hints)[source]: Returns the real number with a zero imaginary part.

im

class diofant.functions.elementary.complexes.im(arg)[source]

Returns imaginary part of expression.

This function performs only elementary analysis and so it will fail to decompose properly more complicated expressions. If completely simplified result is needed then use Basic.as_real_imag() or perform complex expansion on instance of this function.

Examples

>>> im(2*E)
0
>>> re(2*I + 17)
17
>>> im(x*I)
re(x)
>>> im(re(x) + y)
im(y)

as_real_imag(deep=True, **hints)[source]

Return the imaginary part with a zero real part.

Examples

>>> im(2 + 3*I).as_real_imag()
(3, 0)

sign

class diofant.functions.elementary.complexes.sign(arg)[source]

Returns the complex sign of an expression.

For nonzero complex number z is an equivalent of z/abs(z). Else returns zero.

Examples

>>> sign(-1)
-1
>>> sign(0)
0
>>> sign(-3*I)
-I
>>> sign(1 + I)
sign(1 + I)
>>> _.evalf()
0.707106781186548 + 0.707106781186548*I

See also

Abs, conjugate

Abs

class diofant.functions.elementary.complexes.Abs(arg)[source]

Return the absolute value of the argument.

This is an extension of the built-in function abs() to accept symbolic values. If you pass a Diofant expression to the built-in abs(), it will pass it automatically to Abs().

Examples

>>> Abs(-1)
1
>>> x = Symbol('x', real=True)
>>> abs(-x)  # The Python built-in
Abs(x)
>>> abs(x**2)
x**2

Note that the Python built-in will return either an Expr or int depending on the argument:

>>> type(abs(-1))
<... 'int'>
>>> type(abs(Integer(-1)))
<class 'diofant.core.numbers.One'>

Abs will always return a diofant object.

fdiff(argindex=1)[source]

Get the first derivative of the argument to Abs().

Examples

>>> abs(-x).fdiff()
sign(x)

adjoint

class diofant.functions.elementary.complexes.adjoint(arg)[source]: Conjugate transpose or Hermite conjugation.

arg

class diofant.functions.elementary.complexes.arg(arg)[source]

Returns the argument (in radians) of a complex number.

For a real number, the argument is always 0.

Examples

>>> arg(2.0)
0
>>> arg(I)
pi/2
>>> arg(sqrt(2) + I*sqrt(2))
pi/4

conjugate

class diofant.functions.elementary.complexes.conjugate(arg)[source]

Returns the complex conjugate of an argument.

In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part.

Thus, the conjugate of the complex number \(a + i b\) (where a and b are real numbers) is \(a - i b\)

Examples

>>> conjugate(2)
2
>>> conjugate(I)
-I

References

https://en.wikipedia.org/wiki/Complex_conjugation

polar_lift

class diofant.functions.elementary.complexes.polar_lift(arg)[source]

Lift argument to the Riemann surface of the logarithm, using the standard branch.

>>> p = Symbol('p', polar=True)
>>> polar_lift(4)
4*exp_polar(0)
>>> polar_lift(-4)
4*exp_polar(I*pi)
>>> polar_lift(-I)
exp_polar(-I*pi/2)
>>> polar_lift(I + 2)
polar_lift(2 + I)

>>> polar_lift(4*x)
4*polar_lift(x)
>>> polar_lift(4*p)
4*p

periodic_argument

class diofant.functions.elementary.complexes.periodic_argument(ar, period)[source]

Represent the argument on a quotient of the Riemann surface of the logarithm. That is, given a period P, always return a value in (-P/2, P/2], by using exp(P*I) == 1.

>>> unbranched_argument(exp(5*I*pi))
pi
>>> unbranched_argument(exp_polar(5*I*pi))
5*pi
>>> periodic_argument(exp_polar(5*I*pi), 2*pi)
pi
>>> periodic_argument(exp_polar(5*I*pi), 3*pi)
-pi
>>> periodic_argument(exp_polar(5*I*pi), pi)
0

diofant.functions.elementary.complexes.polar_lift: Lift argument to the Riemann surface of the logarithm

diofant.functions.elementary.complexes.principal_branch

principal_branch

class diofant.functions.elementary.complexes.principal_branch(x, period)[source]

Represent a polar number reduced to its principal branch on a quotient of the Riemann surface of the logarithm.

This is a function of two arguments. The first argument is a polar number \(z\), and the second one a positive real number of infinity, \(p\). The result is “z mod exp_polar(I*p)”.

>>> principal_branch(z, oo)
z
>>> principal_branch(exp_polar(2*pi*I)*3, 2*pi)
3*exp_polar(0)
>>> principal_branch(exp_polar(2*pi*I)*3*z, 2*pi)
3*principal_branch(z, 2*pi)

diofant.functions.elementary.complexes.polar_lift: Lift argument to the Riemann surface of the logarithm

diofant.functions.elementary.complexes.periodic_argument

transpose

class diofant.functions.elementary.complexes.transpose(arg)[source]: Linear map transposition.

diofant.functions.elementary.trigonometric

Trigonometric Functions

sin

class diofant.functions.elementary.trigonometric.sin(arg)[source]

The sine function.

Returns the sine of x (measured in radians).

Notes

This function will evaluate automatically in the case x/pi is some rational number. For example, if x is a multiple of pi, pi/2, pi/3, pi/4 and pi/6.

Examples

>>> sin(x**2).diff(x)
2*x*cos(x**2)
>>> sin(pi)
0
>>> sin(pi/2)
1
>>> sin(pi/6)
1/2
>>> sin(pi/12)
-sqrt(2)/4 + sqrt(6)/4

References

cos

class diofant.functions.elementary.trigonometric.cos(arg)[source]

The cosine function.

Returns the cosine of x (measured in radians).

Notes

See sin() for notes about automatic evaluation.

Examples

>>> cos(x**2).diff(x)
-2*x*sin(x**2)
>>> cos(pi)
-1
>>> cos(pi/2)
0
>>> cos(2*pi/3)
-1/2
>>> cos(pi/12)
sqrt(2)/4 + sqrt(6)/4

References

tan

class diofant.functions.elementary.trigonometric.tan(arg)[source]

The tangent function.

Returns the tangent of x (measured in radians).

Notes

See sin() for notes about automatic evaluation.

Examples

>>> tan(x**2).diff(x)
2*x*(tan(x**2)**2 + 1)
>>> tan(pi/8).expand()
-1 + sqrt(2)

References

inverse(argindex=1)[source]: Returns the inverse of this function.

cot

class diofant.functions.elementary.trigonometric.cot(arg)[source]

The cotangent function.

Returns the cotangent of x (measured in radians).

Notes

See sin() for notes about automatic evaluation.

Examples

>>> cot(x**2).diff(x)
2*x*(-cot(x**2)**2 - 1)

References

inverse(argindex=1)[source]: Returns the inverse of this function.

sec

class diofant.functions.elementary.trigonometric.sec(arg)[source]

The secant function.

Returns the secant of x (measured in radians).

Notes

See sin() for notes about automatic evaluation.

Examples

>>> sec(x**2).diff(x)
2*x*tan(x**2)*sec(x**2)

References

csc

class diofant.functions.elementary.trigonometric.csc(arg)[source]

The cosecant function.

Returns the cosecant of x (measured in radians).

Notes

See sin() for notes about automatic evaluation.

Examples

>>> csc(x**2).diff(x)
-2*x*cot(x**2)*csc(x**2)

References

Trigonometric Inverses

asin

class diofant.functions.elementary.trigonometric.asin(arg)[source]

The inverse sine function.

Examples

>>> asin(1)
pi/2
>>> asin(-1)
-pi/2
>>> asin(oo)
-oo*I

See also

sin, csc, acsc

References

inverse(argindex=1)[source]: Returns the inverse of this function.

acos

class diofant.functions.elementary.trigonometric.acos(arg)[source]

The inverse cosine function.

Examples

>>> acos(1)
0
>>> acos(0)
pi/2
>>> acos(oo)
oo*I

See also

cos, sec, asec

References

inverse(argindex=1)[source]: Returns the inverse of this function.

asec

class diofant.functions.elementary.trigonometric.asec(arg)[source]

The inverse secant function.

Examples

>>> asec(1)
0
>>> asec(-1)
pi

See also

sec, cos, acos

References

inverse(argindex=1)[source]: Returns the inverse of this function.

atan

class diofant.functions.elementary.trigonometric.atan(arg)[source]

The inverse tangent function.

Returns the arc tangent of x (measured in radians).

Notes

atan(x) will evaluate automatically in the cases oo, -oo, 0, 1, -1.

Examples

>>> atan(0)
0
>>> atan(1)
pi/4
>>> atan(oo)
pi/2

References

inverse(argindex=1)[source]: Returns the inverse of this function.

acot

class diofant.functions.elementary.trigonometric.acot(arg)[source]

The inverse cotangent function.

Returns the arc cotangent of x (measured in radians). This function has a branch cut discontinuity in the complex plane running from \(-i\) to \(i\).

References

inverse(argindex=1)[source]: Returns the inverse of this function.

acsc

class diofant.functions.elementary.trigonometric.acsc(arg)[source]

The inverse cosecant function.

Examples

>>> acsc(1)
pi/2
>>> acsc(-1)
-pi/2

See also

csc, sin, asin

References

inverse(argindex=1)[source]: Returns the inverse of this function.

atan2

class diofant.functions.elementary.trigonometric.atan2(y, x)[source]

The function atan2(y, x) computes \(\operatorname{atan}(y/x)\) taking two arguments \(y\) and \(x\). Signs of both \(y\) and \(x\) are considered to determine the appropriate quadrant of \(\operatorname{atan}(y/x)\). The range is \((-\pi, \pi]\). The complete definition reads as follows:

\[\begin{split}\operatorname{atan2}(y, x) = \begin{cases} \arctan\left(\frac y x\right) & \qquad x > 0 \\ \arctan\left(\frac y x\right) + \pi& \qquad y \ge 0 , x < 0 \\ \arctan\left(\frac y x\right) - \pi& \qquad y < 0 , x < 0 \\ +\frac{\pi}{2} & \qquad y > 0 , x = 0 \\ -\frac{\pi}{2} & \qquad y < 0 , x = 0 \\ \text{undefined} & \qquad y = 0, x = 0 \end{cases}\end{split}\]

Attention: Note the role reversal of both arguments. The \(y\)-coordinate is the first argument and the \(x\)-coordinate the second.

Examples

Going counter-clock wise around the origin we find the following angles:

>>> atan2(0, 1)
0
>>> atan2(1, 1)
pi/4
>>> atan2(1, 0)
pi/2
>>> atan2(1, -1)
3*pi/4
>>> atan2(0, -1)
pi
>>> atan2(-1, -1)
-3*pi/4
>>> atan2(-1, 0)
-pi/2
>>> atan2(-1, 1)
-pi/4

which are all correct. Compare this to the results of the ordinary \(\operatorname{atan}\) function for the point \((x, y) = (-1, 1)\)

>>> atan(Integer(1) / -1)
-pi/4
>>> atan2(1, -1)
3*pi/4

where only the \(\operatorname{atan2}\) function returns what we expect. We can differentiate the function with respect to both arguments:

>>> diff(atan2(y, x), x)
-y/(x**2 + y**2)

>>> diff(atan2(y, x), y)
x/(x**2 + y**2)

We can express the \(\operatorname{atan2}\) function in terms of complex logarithms:

>>> atan2(y, x).rewrite(log)
-I*log((x + I*y)/sqrt(x**2 + y**2))

and in terms of \(\operatorname(atan)\):

>>> atan2(y, x).rewrite(atan)
2*atan(y/(x + sqrt(x**2 + y**2)))

but note that this form is undefined on the negative real axis.

References

diofant.functions.elementary.hyperbolic

Hyperbolic Functions

HyperbolicFunction

class diofant.functions.elementary.hyperbolic.HyperbolicFunction(*args)[source]: Base class for hyperbolic functions.

See also

diofant.functions.elementary.hyperbolic.sinh, diofant.functions.elementary.hyperbolic.cosh, diofant.functions.elementary.hyperbolic.tanh, diofant.functions.elementary.hyperbolic.coth

sinh

class diofant.functions.elementary.hyperbolic.sinh(arg)[source]

The hyperbolic sine function, \(\frac{e^x - e^{-x}}{2}\).

sinh(x) -> Returns the hyperbolic sine of x

as_real_imag(deep=True, **hints)[source]: Returns this function as a complex coordinate.

fdiff(argindex=1)[source]: Returns the first derivative of this function.

inverse(argindex=1)[source]: Returns the inverse of this function.

static taylor_term(n, x, *previous_terms)[source]: Returns the next term in the Taylor series expansion.

cosh

class diofant.functions.elementary.hyperbolic.cosh(arg)[source]

The hyperbolic cosine function, \(\frac{e^x + e^{-x}}{2}\).

cosh(x) -> Returns the hyperbolic cosine of x

tanh

class diofant.functions.elementary.hyperbolic.tanh(arg)[source]

The hyperbolic tangent function, \(\frac{\sinh(x)}{\cosh(x)}\).

tanh(x) -> Returns the hyperbolic tangent of x

inverse(argindex=1)[source]: Returns the inverse of this function.

coth

class diofant.functions.elementary.hyperbolic.coth(arg)[source]

The hyperbolic cotangent function, \(\frac{\cosh(x)}{\sinh(x)}\).

coth(x) -> Returns the hyperbolic cotangent of x

inverse(argindex=1)[source]: Returns the inverse of this function.

sech

class diofant.functions.elementary.hyperbolic.sech(arg)[source]

The hyperbolic secant function, \(\frac{2}{e^x + e^{-x}}\)

sech(x) -> Returns the hyperbolic secant of x

csch

class diofant.functions.elementary.hyperbolic.csch(arg)[source]

The hyperbolic cosecant function, \(\frac{2}{e^x - e^{-x}}\)

csch(x) -> Returns the hyperbolic cosecant of x

fdiff(argindex=1)[source]: Returns the first derivative of this function.

static taylor_term(n, x, *previous_terms)[source]: Returns the next term in the Taylor series expansion.

Hyperbolic Inverses

asinh

class diofant.functions.elementary.hyperbolic.asinh(arg)[source]

The inverse hyperbolic sine function.

See also

sinh

inverse(argindex=1)[source]: Returns the inverse of this function.

acosh

class diofant.functions.elementary.hyperbolic.acosh(arg)[source]

The inverse hyperbolic cosine function.

See also

cosh

inverse(argindex=1)[source]: Returns the inverse of this function.

atanh

class diofant.functions.elementary.hyperbolic.atanh(arg)[source]

The inverse hyperbolic tangent function.

atanh(x) -> Returns the inverse hyperbolic tangent of x

inverse(argindex=1)[source]: Returns the inverse of this function.

acoth

class diofant.functions.elementary.hyperbolic.acoth(arg)[source]

The inverse hyperbolic cotangent function.

acoth(x) -> Returns the inverse hyperbolic cotangent of x

inverse(argindex=1)[source]: Returns the inverse of this function.

diofant.functions.elementary.integers

ceiling

class diofant.functions.elementary.integers.ceiling(arg)[source]

Ceiling is a univariate function which returns the smallest integer value not less than its argument. Ceiling function is generalized in this implementation to complex numbers.

Examples

>>> ceiling(17)
17
>>> ceiling(Rational(23, 10))
3
>>> ceiling(2*E)
6
>>> ceiling(-Float(0.567))
0
>>> ceiling(I/2)
I

References

“Concrete mathematics” by Graham, pp. 87
https://mathworld.wolfram.com/CeilingFunction.html

floor

class diofant.functions.elementary.integers.floor(arg)[source]

Floor is a univariate function which returns the largest integer value not greater than its argument. However this implementation generalizes floor to complex numbers.

Examples

>>> floor(17)
17
>>> floor(Rational(23, 10))
2
>>> floor(2*E)
5
>>> floor(-Float(0.567))
-1
>>> floor(-I/2)
-I

References

“Concrete mathematics” by Graham, pp. 87
https://mathworld.wolfram.com/FloorFunction.html

RoundFunction

class diofant.functions.elementary.integers.RoundFunction(arg)[source]: The base class for rounding functions.

diofant.functions.elementary.exponential

exp_polar

class diofant.functions.elementary.exponential.exp_polar(*args)[source]

Represent a ‘polar number’ (see g-function Sphinx documentation).

exp_polar represents the function \(Exp: \mathbb{C} \rightarrow \mathcal{S}\), sending the complex number \(z = a + bi\) to the polar number \(r = exp(a), \theta = b\). It is one of the main functions to construct polar numbers.

The main difference is that polar numbers don’t “wrap around” at \(2 \pi\):

>>> exp(2*pi*I)
1
>>> exp_polar(2*pi*I)
exp_polar(2*I*pi)

apart from that they behave mostly like classical complex numbers:

>>> exp_polar(2)*exp_polar(3)
exp_polar(5)

property exp: Returns the exponent of the function.

LambertW

class diofant.functions.elementary.exponential.LambertW(x, k=None)[source]

The Lambert W function \(W(z)\) is defined as the inverse function of \(w \exp(w)\).

In other words, the value of \(W(z)\) is such that \(z = W(z) \exp(W(z))\) for any complex number \(z\). The Lambert W function is a multivalued function with infinitely many branches \(W_k(z)\), indexed by \(k \in \mathbb{Z}\). Each branch gives a different solution \(w\) of the equation \(z = w \exp(w)\).

The Lambert W function has two partially real branches: the principal branch (\(k = 0\)) is real for real \(z > -1/e\), and the \(k = -1\) branch is real for \(-1/e < z < 0\). All branches except \(k = 0\) have a logarithmic singularity at \(z = 0\).

Examples

>>> LambertW(1.2)
0.635564016364870
>>> LambertW(1.2, -1).evalf()
-1.34747534407696 - 4.41624341514535*I
>>> LambertW(-1).is_real
False

References

https://en.wikipedia.org/wiki/Lambert_W_function

fdiff(argindex=1)[source]: Return the first derivative of this function.

diofant.functions.elementary.piecewise

ExprCondPair

class diofant.functions.elementary.piecewise.ExprCondPair(expr, cond)[source]

Represents an expression, condition pair.

property cond: Returns the condition of this pair.

property expr: Returns the expression of this pair.

Piecewise

class diofant.functions.elementary.piecewise.Piecewise(*args)[source]

Represents a piecewise function.

Usage:

Piecewise( (expr,cond), (expr,cond), … )

Each argument is a 2-tuple defining an expression and condition

The conds are evaluated in turn returning the first that is True. If any of the evaluated conds are not determined explicitly False, e.g. x < 1, the function is returned in symbolic form.

Pairs where the cond is explicitly False, will be removed.

The last condition is explicitely True. It’s value is a default value for the function (nan if not specified othewise).

Examples

>>> f = x**2
>>> g = log(x)
>>> p = Piecewise((0, x < -1), (f, x <= 1), (g, True))
>>> p.subs({x: 1})
1
>>> p.subs({x: 5})
log(5)

doit(**hints)[source]: Evaluate this piecewise function.

diofant.functions.elementary.piecewise.piecewise_fold(expr)[source]

Takes an expression containing a piecewise function and returns the expression in piecewise form.

Examples

>>> p = Piecewise((x, x < 1), (1, True))
>>> piecewise_fold(x*p)
Piecewise((x**2, x < 1), (x, true))

diofant.functions.elementary.miscellaneous

IdentityFunction

class diofant.functions.elementary.miscellaneous.IdentityFunction(*args, **kwargs)[source]

The identity function

Examples

>>> Id(x)
x

Min

class diofant.functions.elementary.miscellaneous.Min(*args)[source]

Return, if possible, the minimum value of the list.

It is named Min and not min to avoid conflicts with the built-in function min.

Examples

>>> p = Symbol('p', positive=True)
>>> n = Symbol('n', negative=True)

>>> Min(x, -2)
Min(-2, x)
>>> Min(x, -2).subs({x: 3})
-2
>>> Min(p, -3)
-3
>>> Min(x, y)
Min(x, y)
>>> Min(n, 8, p, -7, p, oo)
Min(-7, n)

See also

diofant.functions.elementary.miscellaneous.Max: find maximum values

Max

class diofant.functions.elementary.miscellaneous.Max(*args)[source]

Return, if possible, the maximum value of the list.

When number of arguments is equal one, then return this argument.

When number of arguments is equal two, then return, if possible, the value from (a, b) that is >= the other.

In common case, when the length of list greater than 2, the task is more complicated. Return only the arguments, which are greater than others, if it is possible to determine directional relation.

If is not possible to determine such a relation, return a partially evaluated result.

Assumptions are used to make the decision too.

Also, only comparable arguments are permitted.

It is named Max and not max to avoid conflicts with the built-in function max.

Examples

>>> p = Symbol('p', positive=True)
>>> n = Symbol('n', negative=True)

>>> Max(x, -2)
Max(-2, x)
>>> Max(x, -2).subs({x: 3})
3
>>> Max(p, -2)
p
>>> Max(x, y)
Max(x, y)
>>> Max(x, y) == Max(y, x)
True
>>> Max(x, Max(y, z))
Max(x, y, z)
>>> Max(n, 8, p, 7, -oo)
Max(8, p)
>>> Max(1, x, oo)
oo

Notes

The task can be considered as searching of supremums in the directed complete partial orders.

The source values are sequentially allocated by the isolated subsets in which supremums are searched and result as Max arguments.

If the resulted supremum is single, then it is returned.

The isolated subsets are the sets of values which are only the comparable with each other in the current set. E.g. natural numbers are comparable with each other, but not comparable with the \(x\) symbol. Another example: the symbol \(x\) with negative assumption is comparable with a natural number.

Also there are “least” elements, which are comparable with all others, and have a zero property (maximum or minimum for all elements). E.g. \(oo\). In case of it the allocation operation is terminated and only this value is returned.

Assumption:

if A > B > C then A > C
if A == B then B can be removed

References

See also

diofant.functions.elementary.miscellaneous.Min: find minimum values

root

diofant.functions.elementary.miscellaneous.root(arg, n, k=0, **kwargs)[source]

Returns the k-th n-th root of arg, defaulting to the principle root.

Examples

>>> root(x, 2)
sqrt(x)

>>> root(x, 3)
root(x, 3)

>>> root(x, n)
x**(1/n)

>>> root(x, -Rational(2, 3))
1/sqrt(x)**3

To get the k-th n-th root, specify k:

>>> root(-2, 3, 2)
-root(-1, 3)**2*root(2, 3)

To get all n n-th roots you can use the RootOf function. The following examples show the roots of unity for n equal 2, 3 and 4:

>>> [RootOf(x**2 - 1, i) for i in range(2)]
[-1, 1]

>>> [RootOf(x**3 - 1, i) for i in range(3)]
[1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2]

>>> [RootOf(x**4 - 1, i) for i in range(4)]
[-1, 1, -I, I]

Diofant, like other symbolic algebra systems, returns the complex root of negative numbers. This is the principal root and differs from the text-book result that one might be expecting. For example, the cube root of -8 does not come back as -2:

>>> root(-8, 3)
2*root(-1, 3)

The real_root function can be used to either make the principle result real (or simply to return the real root directly):

>>> real_root(_)
-2
>>> real_root(-32, 5)
-2

Alternatively, the n//2-th n-th root of a negative number can be computed with root:

>>> root(-32, 5, 5//2)
-2

References

real_root

diofant.functions.elementary.miscellaneous.real_root(arg, n=None)[source]

Return the real nth-root of arg if possible. If n is omitted then all instances of (-n)**(1/odd) will be changed to -n**(1/odd); this will only create a real root of a principle root – the presence of other factors may cause the result to not be real.

Examples

>>> real_root(-8, 3)
-2
>>> root(-8, 3)
2*root(-1, 3)
>>> real_root(_)
-2

If one creates a non-principle root and applies real_root, the result will not be real (so use with caution):

>>> root(-8, 3, 2)
-2*root(-1, 3)**2
>>> real_root(_)
-2*root(-1, 3)**2

sqrt

diofant.functions.elementary.miscellaneous.sqrt(arg, **kwargs)[source]

The square root function

sqrt(x) -> Returns the principal square root of x.

Examples

>>> sqrt(x)
sqrt(x)

>>> sqrt(x)**2
x

Note that sqrt(x**2) does not simplify to x.

>>> sqrt(x**2)
sqrt(x**2)

This is because the two are not equal to each other in general. For example, consider x == -1:

>>> Eq(sqrt(x**2), x).subs({x: -1})
false

This is because sqrt computes the principal square root, so the square may put the argument in a different branch. This identity does hold if x is positive:

>>> y = Symbol('y', positive=True)
>>> sqrt(y**2)
y

You can force this simplification by using the powdenest() function with the force option set to True:

>>> sqrt(x**2)
sqrt(x**2)
>>> powdenest(sqrt(x**2), force=True)
x

To get both branches of the square root you can use the RootOf function:

>>> [RootOf(x**2 - 3, i) for i in (0, 1)]
[-sqrt(3), sqrt(3)]

References