# Refine¶

sympy.assumptions.refine.refine(expr, assumptions=True)[source]

Simplify an expression using assumptions.

Explanation

Unlike `simplify()` which performs structural simplification without any assumption, this function transforms the expression into the form which is only valid under certain assumptions. Note that `simplify()` is generally not done in refining process.

Refining boolean expression involves reducing it to `S.true` or `S.false`. Unlike `ask()`, the expression will not be reduced if the truth value cannot be determined.

Examples

```>>> from sympy import refine, sqrt, Q
>>> from sympy.abc import x
>>> refine(sqrt(x**2), Q.real(x))
Abs(x)
>>> refine(sqrt(x**2), Q.positive(x))
x
```
```>>> refine(Q.real(x), Q.positive(x))
True
>>> refine(Q.positive(x), Q.real(x))
Q.positive(x)
```

`sympy.simplify.simplify.simplify`

Structural simplification without assumptions.

`sympy.assumptions.ask.ask`

Query for boolean expressions using assumptions.

sympy.assumptions.refine.refine_Pow(expr, assumptions)[source]

Handler for instances of Pow.

Examples

```>>> from sympy import Q
>>> from sympy.assumptions.refine import refine_Pow
>>> from sympy.abc import x,y,z
>>> refine_Pow((-1)**x, Q.real(x))
>>> refine_Pow((-1)**x, Q.even(x))
1
>>> refine_Pow((-1)**x, Q.odd(x))
-1
```

For powers of -1, even parts of the exponent can be simplified:

```>>> refine_Pow((-1)**(x+y), Q.even(x))
(-1)**y
>>> refine_Pow((-1)**(x+y+z), Q.odd(x) & Q.odd(z))
(-1)**y
>>> refine_Pow((-1)**(x+y+2), Q.odd(x))
(-1)**(y + 1)
>>> refine_Pow((-1)**(x+3), True)
(-1)**(x + 1)
```
sympy.assumptions.refine.refine_abs(expr, assumptions)[source]

Handler for the absolute value.

Examples

```>>> from sympy import Q, Abs
>>> from sympy.assumptions.refine import refine_abs
>>> from sympy.abc import x
>>> refine_abs(Abs(x), Q.real(x))
>>> refine_abs(Abs(x), Q.positive(x))
x
>>> refine_abs(Abs(x), Q.negative(x))
-x
```
sympy.assumptions.refine.refine_arg(expr, assumptions)[source]

Handler for complex argument

Explanation

```>>> from sympy.assumptions.refine import refine_arg
>>> from sympy import Q, arg
>>> from sympy.abc import x
>>> refine_arg(arg(x), Q.positive(x))
0
>>> refine_arg(arg(x), Q.negative(x))
pi
```
sympy.assumptions.refine.refine_atan2(expr, assumptions)[source]

Handler for the atan2 function.

Examples

```>>> from sympy import Q, atan2
>>> from sympy.assumptions.refine import refine_atan2
>>> from sympy.abc import x, y
>>> refine_atan2(atan2(y,x), Q.real(y) & Q.positive(x))
atan(y/x)
>>> refine_atan2(atan2(y,x), Q.negative(y) & Q.negative(x))
atan(y/x) - pi
>>> refine_atan2(atan2(y,x), Q.positive(y) & Q.negative(x))
atan(y/x) + pi
>>> refine_atan2(atan2(y,x), Q.zero(y) & Q.negative(x))
pi
>>> refine_atan2(atan2(y,x), Q.positive(y) & Q.zero(x))
pi/2
>>> refine_atan2(atan2(y,x), Q.negative(y) & Q.zero(x))
-pi/2
>>> refine_atan2(atan2(y,x), Q.zero(y) & Q.zero(x))
nan
```
sympy.assumptions.refine.refine_im(expr, assumptions)[source]

Handler for imaginary part.

Explanation

```>>> from sympy.assumptions.refine import refine_im
>>> from sympy import Q, im
>>> from sympy.abc import x
>>> refine_im(im(x), Q.real(x))
0
>>> refine_im(im(x), Q.imaginary(x))
-I*x
```
sympy.assumptions.refine.refine_matrixelement(expr, assumptions)[source]

Handler for symmetric part.

Examples

```>>> from sympy.assumptions.refine import refine_matrixelement
>>> from sympy import MatrixSymbol, Q
>>> X = MatrixSymbol('X', 3, 3)
>>> refine_matrixelement(X[0, 1], Q.symmetric(X))
X[0, 1]
>>> refine_matrixelement(X[1, 0], Q.symmetric(X))
X[0, 1]
```
sympy.assumptions.refine.refine_re(expr, assumptions)[source]

Handler for real part.

Examples

```>>> from sympy.assumptions.refine import refine_re
>>> from sympy import Q, re
>>> from sympy.abc import x
>>> refine_re(re(x), Q.real(x))
x
>>> refine_re(re(x), Q.imaginary(x))
0
```
sympy.assumptions.refine.refine_sign(expr, assumptions)[source]

Handler for sign.

Examples

```>>> from sympy.assumptions.refine import refine_sign
>>> from sympy import Symbol, Q, sign, im
>>> x = Symbol('x', real = True)
>>> expr = sign(x)
>>> refine_sign(expr, Q.positive(x) & Q.nonzero(x))
1
>>> refine_sign(expr, Q.negative(x) & Q.nonzero(x))
-1
>>> refine_sign(expr, Q.zero(x))
0
>>> y = Symbol('y', imaginary = True)
>>> expr = sign(y)
>>> refine_sign(expr, Q.positive(im(y)))
I
>>> refine_sign(expr, Q.negative(im(y)))
-I
```