Merge 33266f262f5c45fd9adf9ad3b7d02fb378d0b05b into 338cbafe0d5b07d57f83060ea0f9ba3a6c1155e7

* Fix And Add power.py

To fix the inaccuracies and allow handling of negative exponents and bases, the key issue lies in how negative numbers are handled in the power calculation, especially when dividing.
## Example Output:
```python
>>> power(4, 6)
4096
>>> power(2, 3)
8
>>> power(-2, 3)
-8
>>> power(2, -3)
0.125
>>> power(-2, -3)
-0.125
```

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* Update power.py

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

* Update power.py

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

* Update power.py

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

* Update power.py

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Maxim Smolskiy <mithridatus@mail.ru>

DIRECTORY.md

@@ -726,6 +726,7 @@
   * [Pollard Rho](maths/pollard_rho.py)
   * [Polynomial Evaluation](maths/polynomial_evaluation.py)
   * Polynomials
+    * [Legendre](maths/polynomials/legendre.py)
     * [Single Indeterminate Operations](maths/polynomials/single_indeterminate_operations.py)
   * [Power Using Recursion](maths/power_using_recursion.py)
   * [Prime Check](maths/prime_check.py)

divide_and_conquer/power.py

@@ -1,4 +1,4 @@
-def actual_power(a: int, b: int):
+def actual_power(a: int, b: int) -> int:
     """
     Function using divide and conquer to calculate a^b.
     It only works for integer a,b.
@@ -19,10 +19,12 @@ def actual_power(a: int, b: int):
     """
     if b == 0:
         return 1
+    half = actual_power(a, b // 2)
+
     if (b % 2) == 0:
-        return actual_power(a, int(b / 2)) * actual_power(a, int(b / 2))
+        return half * half
     else:
-        return a * actual_power(a, int(b / 2)) * actual_power(a, int(b / 2))
+        return a * half * half
 
 
 def power(a: int, b: int) -> float:
@@ -43,9 +45,9 @@ def power(a: int, b: int) -> float:
     -0.125
     """
     if b < 0:
-        return 1 / actual_power(a, b)
+        return 1 / actual_power(a, -b)
     return actual_power(a, b)
 
 
 if __name__ == "__main__":
-    print(power(-2, -3))
+    print(power(-2, -3))  # output -0.125

maths/polynomials/legendre.py

@@ -0,0 +1,55 @@
+# Imports de bibliothèques standard
+from math import factorial
+
+# Imports de bibliothèques tierces
+import pytest
+from numpy.polynomial import Polynomial
+
+
+def legendre(n: int) -> list[float]:
+    """
+    Compute the coefficients of the nth Legendre polynomial.
+
+    The Legendre polynomials are solutions to Legendre's differential equation
+    and are widely used in physics and engineering.
+
+    Parameters:
+        n (int): The order of the Legendre polynomial.
+
+    Returns:
+        list[float]: Coefficients of the polynomial in ascending order of powers.
+    """
+    legendre_polynomial = (1 / (factorial(n) * (2**n))) * (Polynomial([-1, 0, 1]) ** n)
+    return legendre_polynomial.deriv(n).coef.tolist()
+
+
+def test_legendre_0() -> None:
+    """Test the 0th Legendre polynomial."""
+    assert legendre(0) == [1.0], "The 0th Legendre polynomial should be [1.0]"
+
+
+def test_legendre_1() -> None:
+    """Test the 1st Legendre polynomial."""
+    assert legendre(1) == [0.0, 1.0], "The 1st Legendre polynomial should be [0.0, 1.0]"
+
+
+def test_legendre_2() -> None:
+    """Test the 2nd Legendre polynomial."""
+    assert legendre(2) == [-0.5, 0.0, 1.5]
+    "The 2nd Legendre polynomial should be [-0.5, 0.0, 1.5]"
+
+
+def test_legendre_3() -> None:
+    """Test the 3rd Legendre polynomial."""
+    assert legendre(3) == [0.0, -1.5, 0.0, 2.5]
+    "The 3rd Legendre polynomial should be [0.0, -1.5, 0.0, 2.5]"
+
+
+def test_legendre_4() -> None:
+    """Test the 4th Legendre polynomial."""
+    assert legendre(4) == pytest.approx([0.375, 0.0, -3.75, 0.0, 4.375])
+    "The 4th Legendre polynomial should be [0.375, 0.0, -3.75, 0.0, 4.375]"
+
+
+if __name__ == "__main__":
+    pytest.main()