Lektion 2

In [1]:

    

from sympy import *
init_printing()


    

In [2]:

    

x = Symbol('x')
y = Symbol('y')
f = (x - y) * (x + y)
f


    

    
Out[2]:





$$\left(x - y\right) \left(x + y\right)$$

In [3]:

    

f.expand()


    

    
Out[3]:





$$x^{2} - y^{2}$$

In [4]:

    

expand(f)


    

    
Out[4]:





$$x^{2} - y^{2}$$

In [5]:

    

expand(f**2)


    

    
Out[5]:





$$x^{4} - 2 x^{2} y^{2} + y^{4}$$

In [6]:

    

fe = f.expand()
fe


    

    
Out[6]:





$$x^{2} - y^{2}$$

In [7]:

    

fe.factor()


    

    
Out[7]:





$$\left(x - y\right) \left(x + y\right)$$

In [8]:

    

g = (x**2-y**2)/(x-y)**2
g


    

    
Out[8]:





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

In [9]:

    

g.ratsimp()


    

    
Out[9]:





$$\frac{2 y}{x - y} + 1$$

In [10]:

    

g.factor()


    

    
Out[10]:





$$\frac{x + y}{x - y}$$

In [11]:

    

g.simplify()


    

    
Out[11]:





$$\frac{x + y}{x - y}$$

In [12]:

    

h = x*x**y
h


    

    
Out[12]:





$$x x^{y}$$

In [13]:

    

h.powsimp()


    

    
Out[13]:





$$x^{y + 1}$$

In [14]:

    

hp = h.simplify()
hp


    

    
Out[14]:





$$x^{y + 1}$$

In [15]:

    

hp.expand()


    

    
Out[15]:





$$x x^{y}$$

In [16]:

    

f = (sin(2*x)+cos(x)) / ((sin(2*x)**2 - cos(x)**2)*(sin(2*x)-cos(x)))
f


    

    
Out[16]:





$$\frac{\sin{\left (2 x \right )} + \cos{\left (x \right )}}{\left(\sin{\left (2 x \right )} - \cos{\left (x \right )}\right) \left(\sin^{2}{\left (2 x \right )} - \cos^{2}{\left (x \right )}\right)}$$

In [17]:

    

f.ratsimp()


    

    
Out[17]:





$$\frac{1}{\sin^{2}{\left (2 x \right )} - 2 \sin{\left (2 x \right )} \cos{\left (x \right )} + \cos^{2}{\left (x \right )}}$$

In [18]:

    

f.trigsimp()


    

    
Out[18]:





$$\frac{1}{\left(2 \sin{\left (x \right )} - 1\right)^{2} \cos^{2}{\left (x \right )}}$$

In [19]:

    

f.simplify()


    

    
Out[19]:





$$\frac{1}{\left(2 \sin{\left (x \right )} - 1\right)^{2} \cos^{2}{\left (x \right )}}$$

In [20]:

    

f.expand()


    

    
Out[20]:





$$\frac{\sin{\left (2 x \right )}}{\sin^{3}{\left (2 x \right )} - \sin^{2}{\left (2 x \right )} \cos{\left (x \right )} - \sin{\left (2 x \right )} \cos^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}} + \frac{\cos{\left (x \right )}}{\sin^{3}{\left (2 x \right )} - \sin^{2}{\left (2 x \right )} \cos{\left (x \right )} - \sin{\left (2 x \right )} \cos^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}}$$

In [21]:

    

f.expand(trig=True)


    

    
Out[21]:





$$\frac{2 \sin{\left (x \right )} \cos{\left (x \right )}}{8 \sin^{3}{\left (x \right )} \cos^{3}{\left (x \right )} - 4 \sin^{2}{\left (x \right )} \cos^{3}{\left (x \right )} - 2 \sin{\left (x \right )} \cos^{3}{\left (x \right )} + \cos^{3}{\left (x \right )}} + \frac{\cos{\left (x \right )}}{8 \sin^{3}{\left (x \right )} \cos^{3}{\left (x \right )} - 4 \sin^{2}{\left (x \right )} \cos^{3}{\left (x \right )} - 2 \sin{\left (x \right )} \cos^{3}{\left (x \right )} + \cos^{3}{\left (x \right )}}$$

In [22]:

    

f.expand(trig=True, numer=True)


    

    
Out[22]:





$$\frac{2 \sin{\left (x \right )} \cos{\left (x \right )} + \cos{\left (x \right )}}{\left(\sin{\left (2 x \right )} - \cos{\left (x \right )}\right) \left(\sin^{2}{\left (2 x \right )} - \cos^{2}{\left (x \right )}\right)}$$

In [23]:

    

fe = f.expand(trig=True,numer=True).expand(trig=True,denom=True)
fe


    

    
Out[23]:





$$\frac{2 \sin{\left (x \right )} \cos{\left (x \right )} + \cos{\left (x \right )}}{8 \sin^{3}{\left (x \right )} \cos^{3}{\left (x \right )} - 4 \sin^{2}{\left (x \right )} \cos^{3}{\left (x \right )} - 2 \sin{\left (x \right )} \cos^{3}{\left (x \right )} + \cos^{3}{\left (x \right )}}$$

In [24]:

    

fe.factor()


    

    
Out[24]:





$$\frac{1}{\left(2 \sin{\left (x \right )} - 1\right)^{2} \cos^{2}{\left (x \right )}}$$

In [25]:

    

f.expand(gibtsnich=True)


    

    
Out[25]:





$$\frac{\sin{\left (2 x \right )}}{\sin^{3}{\left (2 x \right )} - \sin^{2}{\left (2 x \right )} \cos{\left (x \right )} - \sin{\left (2 x \right )} \cos^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}} + \frac{\cos{\left (x \right )}}{\sin^{3}{\left (2 x \right )} - \sin^{2}{\left (2 x \right )} \cos{\left (x \right )} - \sin{\left (2 x \right )} \cos^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}}$$

In [26]:

    

x = Symbol('x')
a = (9*x**2-5)/((x-2)*(x-3))
a


    

    
Out[26]:





$$\frac{9 x^{2} - 5}{\left(x - 3\right) \left(x - 2\right)}$$

In [27]:

    

a.subs(x, 4)


    

    
Out[27]:





$$\frac{139}{2}$$

In [28]:

    

y = Symbol('y')
b = (x-y)**3 / (x+y)
b


    

    
Out[28]:





$$\frac{\left(x - y\right)^{3}}{x + y}$$

In [29]:

    

b.subs(x, 3).subs(y, 0)


    

    
Out[29]:





$$9$$

In [30]:

    

a


    

    
Out[30]:





$$\frac{9 x^{2} - 5}{\left(x - 3\right) \left(x - 2\right)}$$

In [31]:

    

a.subs(x, 2)


    

    
Out[31]:





$$\tilde{\infty}$$

In [32]:

    

c = 1/cos(x*pi/2)
c


    

    
Out[32]:





$$\frac{1}{\cos{\left (\frac{\pi x}{2} \right )}}$$

In [33]:

    

c.subs(x,1)


    

    
Out[33]:





$$\tilde{\infty}$$

In [34]:

    

a


    

    
Out[34]:





$$\frac{9 x^{2} - 5}{\left(x - 3\right) \left(x - 2\right)}$$

In [35]:

    

a.limit(x, 2)


    

    
Out[35]:





$$-\infty$$

In [36]:

    

a.limit(x, 2, dir='-')


    

    
Out[36]:





$$\infty$$

In [37]:

    

a = cos(pi*x)/(2*x-1)
a


    

    
Out[37]:





$$\frac{\cos{\left (\pi x \right )}}{2 x - 1}$$

In [38]:

    

a.limit(x, Rational(1,2))


    

    
Out[38]:





$$- \frac{\pi}{2}$$

In [39]:

    

a.subs(x, Rational(1,2))


    

    
Out[39]:





$$\mathrm{NaN}$$

In [40]:

    

n = Symbol('n')
b = factorial(n)*exp(n)/n**n/sqrt(n)
b


    

    
Out[40]:





$$\frac{n^{- n}}{\sqrt{n}} e^{n} n!$$

In [41]:

    

L = Limit(b, n, oo)   # Träger Operator
L


    

    
Out[41]:





$$\lim_{n \to \infty}\left(\frac{n^{- n}}{\sqrt{n}} e^{n} n!\right)$$

In [42]:

    

L.doit()


    

    
Out[42]:





$$\sqrt{2} \sqrt{\pi}$$

In [43]:

    

f = x**n
f


    

    
Out[43]:





$$x^{n}$$

In [44]:

    

f.diff(x)


    

    
Out[44]:





$$\frac{n x^{n}}{x}$$

In [45]:

    

f.diff(x).powsimp()


    

    
Out[45]:





$$n x^{n - 1}$$

In [46]:

    

f.diff(x,x,x).factor()


    

    
Out[46]:





$$\frac{n x^{n}}{x^{3}} \left(n - 2\right) \left(n - 1\right)$$

In [47]:

    

f.diff(x,3).factor()


    

    
Out[47]:





$$\frac{n x^{n}}{x^{3}} \left(n - 2\right) \left(n - 1\right)$$

In [48]:

    

f.diff()


    

    




---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-48-90cbeb354bfa> in <module>()
----> 1 f.diff()

/az076/anaconda3/lib/python3.5/site-packages/sympy/core/expr.py in diff(self, *symbols, **assumptions)
   2864         new_symbols = list(map(sympify, symbols))  # e.g. x, 2, y, z
   2865         assumptions.setdefault("evaluate", True)
-> 2866         return Derivative(self, *new_symbols, **assumptions)
   2867 
   2868     ###########################################################################

/az076/anaconda3/lib/python3.5/site-packages/sympy/core/function.py in __new__(cls, expr, *variables, **assumptions)
   1035                         Since there is more than one variable in the
   1036                         expression, the variable(s) of differentiation
-> 1037                         must be supplied to differentiate %s''' % expr))
   1038 
   1039         # Standardize the variables by sympifying them and making appending a

ValueError: 
Since there is more than one variable in the expression, the
variable(s) of differentiation must be supplied to differentiate x**n

In [49]:

    

f.diff(x,0)


    

    
Out[49]:





$$x^{n}$$

In [50]:

    

g = x**2*cos(2*x)
g.diff()


    

    
Out[50]:





$$- 2 x^{2} \sin{\left (2 x \right )} + 2 x \cos{\left (2 x \right )}$$

In [51]:

    

a


    

    
Out[51]:





$$\frac{\cos{\left (\pi x \right )}}{2 x - 1}$$

In [52]:

    

zaehler = numer(a)    # numer ist keine Methode
nenner = denom(a)
zaehler, nenner


    

    
Out[52]:





$$\left ( \cos{\left (\pi x \right )}, \quad 2 x - 1\right )$$

In [53]:

    

d_zaehler = zaehler.diff(x)
d_nenner = nenner.diff(x)
d_zaehler, d_nenner


    

    
Out[53]:





$$\left ( - \pi \sin{\left (\pi x \right )}, \quad 2\right )$$

In [54]:

    

d_zaehler.subs(x, Rational(1,2)) / d_nenner.subs(x, Rational(1,2))


    

    
Out[54]:





$$- \frac{\pi}{2}$$

In [55]:

    

f


    

    
Out[55]:





$$x^{n}$$

In [56]:

    

f.integrate(x)


    

    
Out[56]:





$$\begin{cases} \log{\left (x \right )} & \text{for}\: n = -1 \\\frac{x^{n + 1}}{n + 1} & \text{otherwise} \end{cases}$$

In [57]:

    

f.integrate(x, conds='none')


    

    
Out[57]:





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

In [58]:

    

f = sin(2*x)*cos(3*x)**2
f


    

    
Out[58]:





$$\sin{\left (2 x \right )} \cos^{2}{\left (3 x \right )}$$

In [59]:

    

I1 = f.integrate()   # warum darf diese Variable nicht I heißen?
I1


    

    
Out[59]:





$$\frac{3}{16} \sin{\left (2 x \right )} \sin{\left (3 x \right )} \cos{\left (3 x \right )} - \frac{9}{32} \sin^{2}{\left (3 x \right )} \cos{\left (2 x \right )} - \frac{7}{32} \cos{\left (2 x \right )} \cos^{2}{\left (3 x \right )}$$

In [60]:

    

I1.diff(x)


    

    
Out[60]:





$$\sin{\left (2 x \right )} \cos^{2}{\left (3 x \right )}$$

In [61]:

    

I1.diff(x) == f


    

    
Out[61]:





True

In [62]:

    

g = 1/(1+x**4)
I2 = g.integrate(x)
I2


    

    
Out[62]:





$$- \frac{\sqrt{2}}{8} \log{\left (x^{2} - \sqrt{2} x + 1 \right )} + \frac{\sqrt{2}}{8} \log{\left (x^{2} + \sqrt{2} x + 1 \right )} + \frac{\sqrt{2}}{4} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )} + \frac{\sqrt{2}}{4} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}$$

In [63]:

    

I2.diff(x)


    

    
Out[63]:





$$- \frac{\sqrt{2} \left(2 x - \sqrt{2}\right)}{8 x^{2} - 8 \sqrt{2} x + 8} + \frac{\sqrt{2} \left(2 x + \sqrt{2}\right)}{8 x^{2} + 8 \sqrt{2} x + 8} + \frac{1}{2 \left(\sqrt{2} x + 1\right)^{2} + 2} + \frac{1}{2 \left(\sqrt{2} x - 1\right)^{2} + 2}$$

In [64]:

    

I2.diff() == g


    

    
Out[64]:





False

In [65]:

    

I2.diff().ratsimp() ==g


    

    
Out[65]:





True

In [66]:

    

f


    

    
Out[66]:





$$\sin{\left (2 x \right )} \cos^{2}{\left (3 x \right )}$$

In [67]:

    

I3 = f.integrate((x,0,pi/4))
I3


    

    
Out[67]:





$$\frac{1}{8}$$

In [68]:

    

I4 = Integral(f, (x,0,pi/4))
I4


    

    
Out[68]:





$$\int_{0}^{\frac{\pi}{4}} \sin{\left (2 x \right )} \cos^{2}{\left (3 x \right )}\, dx$$

In [69]:

    

I4.doit()


    

    
Out[69]:





$$\frac{1}{8}$$

In [70]:

    

I4.n()  # numerische Integration


    

    
Out[70]:





$$0.125$$

In [71]:

    

I5 = Integral(sqrt(exp(x)+4), (x,0,1))
I5


    

    
Out[71]:





$$\int_{0}^{1} \sqrt{e^{x} + 4}\, dx$$

In [72]:

    

I5.doit()


    

    
Out[72]:





$$\int_{0}^{1} \sqrt{e^{x} + 4}\, dx$$

In [73]:

    

I5.n()


    

    
Out[73]:





$$2.38910363799198$$

In [74]:

    

import numpy as np
from scipy import integrate


    

In [75]:

    

def f(x):
    return np.sqrt(np.exp(x)+4)


    

In [76]:

    

res, err = integrate.quad(f, 0, 1)
res


    

    
Out[76]:





$$2.3891036379919814$$