Simplification Solutions



In [1]:

    
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.



In [2]:

    
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
    """
    return collect(expr, cos(x))



In [3]:

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









    Out[3]:





$$\left(\sin{\left (x \right )} + 1\right) \cos{\left (x \right )}$$



In [4]:

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









    Out[4]:





$$\left(\sin{\left (x \right )} + 1\right) \cos{\left (x \right )} + 1$$



In [5]:

    
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
    """
    return expand(apart(expr))



In [6]:

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









    Out[6]:





$$\frac{1}{x + 1} + \frac{1}{x}$$



In [7]:

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









    Out[7]:





$$\frac{1}{x^{2} + 2 x + 1} + \frac{1}{x}$$

Powers

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



In [8]:

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



In [9]:

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









    Out[9]:





$$e^{x} + e^{x + y}$$



In [10]:

    
def powersimp2(expr):
    """
    >>> powersimp2(2**x*x**x)
    (2*x)**x
    >>> powersimp2(x**x*x**x)
    (x**2)**x
    """
    return powsimp(expr, force=True)



In [11]:

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









    Out[11]:





$$\left(2 x\right)^{x}$$



In [12]:

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









    Out[12]:





$$\left(x^{2}\right)^{x}$$



In [13]:

    
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)
    """
    return powdenest(expand_power_exp(expr), force=True)



In [14]:

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



In [15]:

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









    Out[15]:





$$a^{b c}$$



In [16]:

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









    Out[16]:





$$a^{b c + b}$$

Logs



In [17]:

    
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)
    """
    return logcombine(expand_log(expr, force=True))



In [18]:

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



In [19]:

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









    Out[19]:





$$y \log{\left (x \right )} + \log{\left (a^{b} \right )}$$



In [20]:

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









    Out[20]:





$$\log{\left (x \right )} + \log{\left (y \right )} + \log{\left (a b \right )}$$

Miscellaneous



In [21]:

    
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))
    """
    return expand_trig(expr).rewrite(tan)



In [22]:

    
miscsimp1(sin(x + y))









    Out[22]:





$$\frac{2 \left(- \tan^{2}{\left (\frac{x}{2} \right )} + 1\right) \tan{\left (\frac{y}{2} \right )}}{\left(\tan^{2}{\left (\frac{x}{2} \right )} + 1\right) \left(\tan^{2}{\left (\frac{y}{2} \right )} + 1\right)} + \frac{2 \left(- \tan^{2}{\left (\frac{y}{2} \right )} + 1\right) \tan{\left (\frac{x}{2} \right )}}{\left(\tan^{2}{\left (\frac{x}{2} \right )} + 1\right) \left(\tan^{2}{\left (\frac{y}{2} \right )} + 1\right)}$$



In [23]:

    
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)
    """
    return expand(expand_func(expr))



In [24]:

    
miscsimp2(gamma(x + 4))









    Out[24]:





$$x^{4} \Gamma{\left(x \right)} + 6 x^{3} \Gamma{\left(x \right)} + 11 x^{2} \Gamma{\left(x \right)} + 6 x \Gamma{\left(x \right)}$$

Continued Fractions

If we do not cover this, see http://asmeurer.github.io/scipy-2014-tutorial/html/tutorial/simplification.html#example-continued-fractions.



In [25]:

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



In [26]:

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

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



In [28]:

    
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
    """
    return [a3, a4, a0, a2, a1]



In [29]:

    
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)



In [30]:

    
orig_frac









    Out[30]:





$$\frac{a_{0} a_{1} a_{2} a_{3} a_{4} + a_{0} a_{1} a_{2} + a_{0} a_{3} a_{4} + a_{0} + a_{1} a_{2} a_{3} + a_{1} a_{3} a_{4} + a_{1} + a_{3}}{a_{0} a_{1} a_{2} a_{4} + a_{0} a_{4} + a_{1} a_{2} + a_{1} a_{4} + 1}$$



In [31]:

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









    Out[31]:





True



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