# Simplification¶

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from sympy import *
x, y, z = symbols('x y z')
init_printing()


For each exercise, fill in the function according to its docstring.

## Polynomial/Rational Function Simplification¶

In each exercise, apply specific simplification functions to get the desired result.

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def polysimp1(expr):
"""
>>> polysimp1(cos(x)*sin(x) + cos(x))
(sin(x) + 1)*cos(x)
>>> polysimp1(cos(x)*sin(x) + cos(x) + 1)
(sin(x) + 1)*cos(x) + 1
"""

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polysimp1(cos(x)*sin(x) + cos(x))

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polysimp1(cos(x)*sin(x) + cos(x) + 1)

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def polysimp2(expr):
"""
>>> polysimp2((2*x + 1)/(x**2 + x))
1/(x + 1) + 1/x
>>> polysimp2((x**2 + 3*x + 1)/(x**3 + 2*x**2 + x))
1/(x**2 + 2*x + 1) + 1/x
"""

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polysimp2((2*x + 1)/(x**2 + x))

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polysimp2((x**2 + 3*x + 1)/(x**3 + 2*x**2 + x))


## Powers¶

In each exercise, apply specific simplification functions to get the desired result.

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def powersimp1(expr):
"""
>>> powersimp1(exp(x)*(exp(y) + 1))
exp(x) + exp(x + y)
"""

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powersimp1(exp(x)*(exp(y) + 1))

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def powersimp2(expr):
"""
>>> powersimp2(2**x*x**x)
(2*x)**x
>>> powersimp2(x**x*x**x)
(x**2)**x
"""

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powersimp2(2**x*x**x)

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powersimp2(x**x*x**x)

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def powersimp3(expr):
"""
>>> a, b, c = symbols('a b c')
>>> powersimp3((a**b)**c)
a**(b*c)
>>> powersimp3((a**b)**(c + 1))
a**(b*c + b)
"""

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a, b, c = symbols('a b c')

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powersimp3((a**b)**c)

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powersimp3((a**b)**(c + 1))


## Logs¶

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def logsimp1(expr):
"""
>>> a, b = symbols('a b', positive=True)
>>> logsimp1(log(x**y*a**b))
y*log(x) + log(a**b)
>>> logsimp1(log(x*y*a*b))
log(x) + log(y) + log(a*b)
"""

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a, b = symbols('a b', positive=True)

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logsimp1(log(x**y*a**b))

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logsimp1(log(x*y*a*b))


## Miscellaneous¶

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def miscsimp1(expr):
"""
>>> miscsimp1(sin(x + y))
2*(-tan(x/2)**2 + 1)*tan(y/2)/((tan(x/2)**2 + 1)*(tan(y/2)**2 + 1)) + 2*(-tan(y/2)**2 + 1)*tan(x/2)/((tan(x/2)**2 + 1)*(tan(y/2)**2 + 1))
"""

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miscsimp1(sin(x + y))

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def miscsimp2(expr):
"""
>>> miscsimp2(gamma(x + 4))
x**4*gamma(x) + 6*x**3*gamma(x) + 11*x**2*gamma(x) + 6*x*gamma(x)
"""

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miscsimp2(gamma(x + 4))


## Continued Fractions¶

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def list_to_frac(l):
expr = Integer(0)
for i in reversed(l[1:]):
expr += i
expr = 1/expr
return l[0] + expr

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a0, a1, a2, a3, a4 = symbols('a0:5')


Determine the list used to create the continued fraction $$\frac{a_{0} a_{1} a_{2} a_{3} a_{4} + a_{0} a_{1} a_{2} + a_{0} a_{3} a_{4} + a_{0} + a_{1} a_{2} a_{3} + a_{1} a_{3} a_{4} + a_{1} + a_{3}}{a_{0} a_{1} a_{2} a_{4} + a_{0} a_{4} + a_{1} a_{2} + a_{1} a_{4} + 1}.$$

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def continued_frac():
"""
Determine the original list used to create the fraction.

Return the original list from this function.

>>> orig_frac = (a0*a1*a2*a3*a4 + a0*a1*a2 + a0*a3*a4 + a0 + a1*a2*a3 + a1*a3*a4 + a1 + a3)/(a0*a1*a2*a4 + a0*a4 + a1*a2 + a1*a4 + 1)
>>> pprint(orig_frac)
a₀⋅a₁⋅a₂⋅a₃⋅a₄ + a₀⋅a₁⋅a₂ + a₀⋅a₃⋅a₄ + a₀ + a₁⋅a₂⋅a₃ + a₁⋅a₃⋅a₄ + a₁ + a₃
─────────────────────────────────────────────────────────────────────────
a₀⋅a₁⋅a₂⋅a₄ + a₀⋅a₄ + a₁⋅a₂ + a₁⋅a₄ + 1
>>> cancel(list_to_frac(continued_frac())) == orig_frac
True
"""

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orig_frac = (a0*a1*a2*a3*a4 + a0*a1*a2 + a0*a3*a4 + a0 + a1*a2*a3 + a1*a3*a4 + a1 + a3)/(a0*a1*a2*a4 + a0*a4 + a1*a2 + a1*a4 + 1)

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orig_frac

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cancel(list_to_frac(continued_frac())) == orig_frac

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