Initial commit

skills/sympy/references/advanced-topics.md

@@ -0,0 +1,635 @@
+# SymPy Advanced Topics
+
+This document covers SymPy's advanced mathematical capabilities including geometry, number theory, combinatorics, logic and sets, statistics, polynomials, and special functions.
+
+## Geometry
+
+### 2D Geometry
+
+```python
+from sympy.geometry import Point, Line, Circle, Triangle, Polygon
+
+# Points
+p1 = Point(0, 0)
+p2 = Point(1, 1)
+p3 = Point(1, 0)
+
+# Distance between points
+dist = p1.distance(p2)
+
+# Lines
+line = Line(p1, p2)
+line_from_eq = Line(Point(0, 0), slope=2)
+
+# Line properties
+line.slope       # Slope
+line.equation()  # Equation of line
+line.length      # oo (infinite for lines)
+
+# Line segment
+from sympy.geometry import Segment
+seg = Segment(p1, p2)
+seg.length       # Finite length
+seg.midpoint     # Midpoint
+
+# Intersection
+line2 = Line(Point(0, 1), Point(1, 0))
+intersection = line.intersection(line2)  # [Point(1/2, 1/2)]
+
+# Circles
+circle = Circle(Point(0, 0), 5)  # Center, radius
+circle.area           # 25*pi
+circle.circumference  # 10*pi
+
+# Triangles
+tri = Triangle(p1, p2, p3)
+tri.area       # Area
+tri.perimeter  # Perimeter
+tri.angles     # Dictionary of angles
+tri.vertices   # Tuple of vertices
+
+# Polygons
+poly = Polygon(Point(0, 0), Point(1, 0), Point(1, 1), Point(0, 1))
+poly.area
+poly.perimeter
+poly.vertices
+```
+
+### Geometric Queries
+
+```python
+# Check if point is on line/curve
+point = Point(0.5, 0.5)
+line.contains(point)
+
+# Check if parallel/perpendicular
+line1 = Line(Point(0, 0), Point(1, 1))
+line2 = Line(Point(0, 1), Point(1, 2))
+line1.is_parallel(line2)  # True
+line1.is_perpendicular(line2)  # False
+
+# Tangent lines
+from sympy.geometry import Circle, Point
+circle = Circle(Point(0, 0), 5)
+point = Point(5, 0)
+tangents = circle.tangent_lines(point)
+```
+
+### 3D Geometry
+
+```python
+from sympy.geometry import Point3D, Line3D, Plane
+
+# 3D Points
+p1 = Point3D(0, 0, 0)
+p2 = Point3D(1, 1, 1)
+p3 = Point3D(1, 0, 0)
+
+# 3D Lines
+line = Line3D(p1, p2)
+
+# Planes
+plane = Plane(p1, p2, p3)  # From 3 points
+plane = Plane(Point3D(0, 0, 0), normal_vector=(1, 0, 0))  # From point and normal
+
+# Plane equation
+plane.equation()
+
+# Distance from point to plane
+point = Point3D(2, 3, 4)
+dist = plane.distance(point)
+
+# Intersection of plane and line
+intersection = plane.intersection(line)
+```
+
+### Curves and Ellipses
+
+```python
+from sympy.geometry import Ellipse, Curve
+from sympy import sin, cos, pi
+
+# Ellipse
+ellipse = Ellipse(Point(0, 0), hradius=3, vradius=2)
+ellipse.area          # 6*pi
+ellipse.eccentricity  # Eccentricity
+
+# Parametric curves
+from sympy.abc import t
+curve = Curve((cos(t), sin(t)), (t, 0, 2*pi))  # Circle
+```
+
+## Number Theory
+
+### Prime Numbers
+
+```python
+from sympy.ntheory import isprime, primerange, prime, nextprime, prevprime
+
+# Check if prime
+isprime(7)    # True
+isprime(10)   # False
+
+# Generate primes in range
+list(primerange(10, 50))  # [11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
+
+# nth prime
+prime(10)     # 29 (10th prime)
+
+# Next and previous primes
+nextprime(10)  # 11
+prevprime(10)  # 7
+```
+
+### Prime Factorization
+
+```python
+from sympy import factorint, primefactors, divisors
+
+# Prime factorization
+factorint(60)  # {2: 2, 3: 1, 5: 1} means 2^2 * 3^1 * 5^1
+
+# List of prime factors
+primefactors(60)  # [2, 3, 5]
+
+# All divisors
+divisors(60)  # [1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60]
+```
+
+### GCD and LCM
+
+```python
+from sympy import gcd, lcm, igcd, ilcm
+
+# Greatest common divisor
+gcd(60, 48)   # 12
+igcd(60, 48)  # 12 (integer version)
+
+# Least common multiple
+lcm(60, 48)   # 240
+ilcm(60, 48)  # 240 (integer version)
+
+# Multiple arguments
+gcd(60, 48, 36)  # 12
+```
+
+### Modular Arithmetic
+
+```python
+from sympy.ntheory import mod_inverse, totient, is_primitive_root
+
+# Modular inverse (find x such that a*x ≡ 1 (mod m))
+mod_inverse(3, 7)  # 5 (because 3*5 = 15 ≡ 1 (mod 7))
+
+# Euler's totient function
+totient(10)  # 4 (numbers less than 10 coprime to 10: 1,3,7,9)
+
+# Primitive roots
+is_primitive_root(2, 5)  # True
+```
+
+### Diophantine Equations
+
+```python
+from sympy.solvers.diophantine import diophantine
+from sympy.abc import x, y, z
+
+# Linear Diophantine: ax + by = c
+diophantine(3*x + 4*y - 5)  # {(4*t_0 - 5, -3*t_0 + 5)}
+
+# Quadratic forms
+diophantine(x**2 + y**2 - 25)  # Pythagorean-type equations
+
+# More complex equations
+diophantine(x**2 - 4*x*y + 8*y**2 - 3*x + 7*y - 5)
+```
+
+### Continued Fractions
+
+```python
+from sympy import nsimplify, continued_fraction_iterator
+from sympy import Rational, pi
+
+# Convert to continued fraction
+cf = continued_fraction_iterator(Rational(415, 93))
+list(cf)  # [4, 2, 6, 7]
+
+# Approximate irrational numbers
+cf_pi = continued_fraction_iterator(pi.evalf(20))
+```
+
+## Combinatorics
+
+### Permutations and Combinations
+
+```python
+from sympy import factorial, binomial, factorial2
+from sympy.functions.combinatorial.numbers import nC, nP
+
+# Factorial
+factorial(5)  # 120
+
+# Binomial coefficient (n choose k)
+binomial(5, 2)  # 10
+
+# Permutations nPk = n!/(n-k)!
+nP(5, 2)  # 20
+
+# Combinations nCk = n!/(k!(n-k)!)
+nC(5, 2)  # 10
+
+# Double factorial n!!
+factorial2(5)  # 15 (5*3*1)
+factorial2(6)  # 48 (6*4*2)
+```
+
+### Permutation Objects
+
+```python
+from sympy.combinatorics import Permutation
+
+# Create permutation (cycle notation)
+p = Permutation([1, 2, 0, 3])  # Sends 0->1, 1->2, 2->0, 3->3
+p = Permutation(0, 1, 2)(3)    # Cycle notation: (0 1 2)(3)
+
+# Permutation operations
+p.order()       # Order of permutation
+p.is_even       # True if even permutation
+p.inversions()  # Number of inversions
+
+# Compose permutations
+q = Permutation([2, 0, 1, 3])
+r = p * q       # Composition
+```
+
+### Partitions
+
+```python
+from sympy.utilities.iterables import partitions
+from sympy.functions.combinatorial.numbers import partition
+
+# Number of integer partitions
+partition(5)  # 7 (5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1)
+
+# Generate all partitions
+list(partitions(4))
+# {4: 1}, {3: 1, 1: 1}, {2: 2}, {2: 1, 1: 2}, {1: 4}
+```
+
+### Catalan and Fibonacci Numbers
+
+```python
+from sympy import catalan, fibonacci, lucas
+
+# Catalan numbers
+catalan(5)  # 42
+
+# Fibonacci numbers
+fibonacci(10)  # 55
+lucas(10)      # 123 (Lucas numbers)
+```
+
+### Group Theory
+
+```python
+from sympy.combinatorics import PermutationGroup, Permutation
+
+# Create permutation group
+p1 = Permutation([1, 0, 2])
+p2 = Permutation([0, 2, 1])
+G = PermutationGroup(p1, p2)
+
+# Group properties
+G.order()        # Order of group
+G.is_abelian     # Check if abelian
+G.is_cyclic()    # Check if cyclic
+G.elements       # All group elements
+```
+
+## Logic and Sets
+
+### Boolean Logic
+
+```python
+from sympy import symbols, And, Or, Not, Xor, Implies, Equivalent
+from sympy.logic.boolalg import truth_table, simplify_logic
+
+# Define boolean variables
+x, y, z = symbols('x y z', bool=True)
+
+# Logical operations
+expr = And(x, Or(y, Not(z)))
+expr = Implies(x, y)  # x -> y
+expr = Equivalent(x, y)  # x <-> y
+expr = Xor(x, y)  # Exclusive OR
+
+# Simplification
+expr = (x & y) | (x & ~y)
+simplified = simplify_logic(expr)  # Returns x
+
+# Truth table
+expr = Implies(x, y)
+print(truth_table(expr, [x, y]))
+```
+
+### Sets
+
+```python
+from sympy import FiniteSet, Interval, Union, Intersection, Complement
+from sympy import S  # For special sets
+
+# Finite sets
+A = FiniteSet(1, 2, 3, 4)
+B = FiniteSet(3, 4, 5, 6)
+
+# Set operations
+union = Union(A, B)              # {1, 2, 3, 4, 5, 6}
+intersection = Intersection(A, B)  # {3, 4}
+difference = Complement(A, B)     # {1, 2}
+
+# Intervals
+I = Interval(0, 1)              # [0, 1]
+I_open = Interval.open(0, 1)    # (0, 1)
+I_lopen = Interval.Lopen(0, 1)  # (0, 1]
+I_ropen = Interval.Ropen(0, 1)  # [0, 1)
+
+# Special sets
+S.Reals        # All real numbers
+S.Integers     # All integers
+S.Naturals     # Natural numbers
+S.EmptySet     # Empty set
+S.Complexes    # Complex numbers
+
+# Set membership
+3 in A  # True
+7 in A  # False
+
+# Subset and superset
+A.is_subset(B)    # False
+A.is_superset(B)  # False
+```
+
+### Set Theory Operations
+
+```python
+from sympy import ImageSet, Lambda
+from sympy.abc import x
+
+# Image set (set of function values)
+squares = ImageSet(Lambda(x, x**2), S.Integers)
+# {x^2 | x ∈ ℤ}
+
+# Power set
+from sympy.sets import FiniteSet
+A = FiniteSet(1, 2, 3)
+# Note: SymPy doesn't have direct powerset, but can generate
+```
+
+## Polynomials
+
+### Polynomial Manipulation
+
+```python
+from sympy import Poly, symbols, factor, expand, roots
+x, y = symbols('x y')
+
+# Create polynomial
+p = Poly(x**2 + 2*x + 1, x)
+
+# Polynomial properties
+p.degree()       # 2
+p.coeffs()       # [1, 2, 1]
+p.as_expr()      # Convert back to expression
+
+# Arithmetic
+p1 = Poly(x**2 + 1, x)
+p2 = Poly(x + 1, x)
+p3 = p1 + p2
+p4 = p1 * p2
+q, r = div(p1, p2)  # Quotient and remainder
+```
+
+### Polynomial Roots
+
+```python
+from sympy import roots, real_roots, count_roots
+
+p = Poly(x**3 - 6*x**2 + 11*x - 6, x)
+
+# All roots
+r = roots(p)  # {1: 1, 2: 1, 3: 1}
+
+# Real roots only
+r = real_roots(p)
+
+# Count roots in interval
+count_roots(p, a, b)  # Number of roots in [a, b]
+```
+
+### Polynomial GCD and Factorization
+
+```python
+from sympy import gcd, lcm, factor, factor_list
+
+p1 = Poly(x**2 - 1, x)
+p2 = Poly(x**2 - 2*x + 1, x)
+
+# GCD and LCM
+g = gcd(p1, p2)
+l = lcm(p1, p2)
+
+# Factorization
+f = factor(x**3 - x**2 + x - 1)  # (x - 1)*(x**2 + 1)
+factors = factor_list(x**3 - x**2 + x - 1)  # List form
+```
+
+### Groebner Bases
+
+```python
+from sympy import groebner, symbols
+
+x, y, z = symbols('x y z')
+polynomials = [x**2 + y**2 + z**2 - 1, x*y - z]
+
+# Compute Groebner basis
+gb = groebner(polynomials, x, y, z)
+```
+
+## Statistics
+
+### Random Variables
+
+```python
+from sympy.stats import (
+    Normal, Uniform, Exponential, Poisson, Binomial,
+    P, E, variance, density, sample
+)
+
+# Define random variables
+X = Normal('X', 0, 1)  # Normal(mean, std)
+Y = Uniform('Y', 0, 1)  # Uniform(a, b)
+Z = Exponential('Z', 1)  # Exponential(rate)
+
+# Probability
+P(X > 0)  # 1/2
+P((X > 0) & (X < 1))
+
+# Expected value
+E(X)  # 0
+E(X**2)  # 1
+
+# Variance
+variance(X)  # 1
+
+# Density function
+density(X)(x)  # sqrt(2)*exp(-x**2/2)/(2*sqrt(pi))
+```
+
+### Discrete Distributions
+
+```python
+from sympy.stats import Die, Bernoulli, Binomial, Poisson
+
+# Die
+D = Die('D', 6)
+P(D > 3)  # 1/2
+
+# Bernoulli
+B = Bernoulli('B', 0.5)
+P(B)  # 1/2
+
+# Binomial
+X = Binomial('X', 10, 0.5)
+P(X == 5)  # Probability of exactly 5 successes in 10 trials
+
+# Poisson
+Y = Poisson('Y', 3)
+P(Y < 2)  # Probability of less than 2 events
+```
+
+### Joint Distributions
+
+```python
+from sympy.stats import Normal, P, E
+from sympy import symbols
+
+# Independent random variables
+X = Normal('X', 0, 1)
+Y = Normal('Y', 0, 1)
+
+# Joint probability
+P((X > 0) & (Y > 0))  # 1/4
+
+# Covariance
+from sympy.stats import covariance
+covariance(X, Y)  # 0 (independent)
+```
+
+## Special Functions
+
+### Common Special Functions
+
+```python
+from sympy import (
+    gamma,      # Gamma function
+    beta,       # Beta function
+    erf,        # Error function
+    besselj,    # Bessel function of first kind
+    bessely,    # Bessel function of second kind
+    hermite,    # Hermite polynomial
+    legendre,   # Legendre polynomial
+    laguerre,   # Laguerre polynomial
+    chebyshevt, # Chebyshev polynomial (first kind)
+    zeta        # Riemann zeta function
+)
+
+# Gamma function
+gamma(5)  # 24 (equivalent to 4!)
+gamma(1/2)  # sqrt(pi)
+
+# Bessel functions
+besselj(0, x)  # J_0(x)
+bessely(1, x)  # Y_1(x)
+
+# Orthogonal polynomials
+hermite(3, x)    # 8*x**3 - 12*x
+legendre(2, x)   # (3*x**2 - 1)/2
+laguerre(2, x)   # x**2/2 - 2*x + 1
+chebyshevt(3, x) # 4*x**3 - 3*x
+```
+
+### Hypergeometric Functions
+
+```python
+from sympy import hyper, meijerg
+
+# Hypergeometric function
+hyper([1, 2], [3], x)
+
+# Meijer G-function
+meijerg([[1, 1], []], [[1], [0]], x)
+```
+
+## Common Patterns
+
+### Pattern 1: Symbolic Geometry Problem
+
+```python
+from sympy.geometry import Point, Triangle
+from sympy import symbols
+
+# Define symbolic triangle
+a, b = symbols('a b', positive=True)
+tri = Triangle(Point(0, 0), Point(a, 0), Point(0, b))
+
+# Compute properties symbolically
+area = tri.area  # a*b/2
+perimeter = tri.perimeter  # a + b + sqrt(a**2 + b**2)
+```
+
+### Pattern 2: Number Theory Calculation
+
+```python
+from sympy.ntheory import factorint, totient, isprime
+
+# Factor and analyze
+n = 12345
+factors = factorint(n)
+phi = totient(n)
+is_prime = isprime(n)
+```
+
+### Pattern 3: Combinatorial Generation
+
+```python
+from sympy.utilities.iterables import multiset_permutations, combinations
+
+# Generate all permutations
+perms = list(multiset_permutations([1, 2, 3]))
+
+# Generate combinations
+combs = list(combinations([1, 2, 3, 4], 2))
+```
+
+### Pattern 4: Probability Calculation
+
+```python
+from sympy.stats import Normal, P, E, variance
+
+X = Normal('X', mu, sigma)
+
+# Compute statistics
+mean = E(X)
+var = variance(X)
+prob = P(X > a)
+```
+
+## Important Notes
+
+1. **Assumptions:** Many operations benefit from symbol assumptions (e.g., `positive=True`, `integer=True`).
+
+2. **Symbolic vs Numeric:** These operations are symbolic. Use `evalf()` for numerical results.
+
+3. **Performance:** Complex symbolic operations can be slow. Consider numerical methods for large-scale computations.
+
+4. **Exact arithmetic:** SymPy maintains exact representations (e.g., `sqrt(2)` instead of `1.414...`).