Lektion 3

In [1]:

    

from sympy import *
init_printing()


    

In [2]:

    

x = Symbol('x')
a = Symbol('a')


    

In [3]:

    

I1 = Integral(x*exp(-a*x), (x,0,oo))
I1


    

    
Out[3]:





$$\int_{0}^{\infty} x e^{- a x}\, dx$$

In [4]:

    

I1.doit()


    

    
Out[4]:





$$\begin{cases} \frac{1}{a^{2}} & \text{for}\: \left|{\operatorname{periodic_{argument}}{\left (a,\infty \right )}}\right| < \frac{\pi}{2} \\\int_{0}^{\infty} x e^{- a x}\, dx & \text{otherwise} \end{cases}$$

In [5]:

    

a = Symbol('a', positive=True)


    

In [6]:

    

sqrt(a**2)


    

    
Out[6]:





$$a$$

In [7]:

    

I1.doit()


    

    
Out[7]:





$$\begin{cases} \frac{1}{a^{2}} & \text{for}\: \left|{\operatorname{periodic_{argument}}{\left (a,\infty \right )}}\right| < \frac{\pi}{2} \\\int_{0}^{\infty} x e^{- a x}\, dx & \text{otherwise} \end{cases}$$

In [8]:

    

I2 = Integral(x*exp(-a*x), (x,0,oo))  # neues I1
I2.doit()


    

    
Out[8]:





$$\frac{1}{a^{2}}$$

In [9]:

    

a._assumptions


    

    
Out[9]:





{'algebraic': None,
 'antihermitian': None,
 'commutative': True,
 'complex': True,
 'finite': None,
 'hermitian': True,
 'imaginary': False,
 'infinite': None,
 'irrational': None,
 'negative': False,
 'nonnegative': True,
 'nonpositive': False,
 'nonzero': True,
 'polar': None,
 'positive': True,
 'rational': None,
 'real': True,
 'zero': False}

In [10]:

    

n = Symbol('n', integer=True)
cos(pi*n)


    

    
Out[10]:





$$\left(-1\right)^{n}$$

In [11]:

    

n = Symbol('n')
j = Symbol('j')
q = Symbol('q')


    

In [12]:

    

S1 = Sum(j, (j,1,n))   # Sum gibt es nur als trägen Operator
S1


    

    
Out[12]:





$$\sum_{j=1}^{n} j$$

In [13]:

    

S1.doit().factor()


    

    
Out[13]:





$$\frac{n}{2} \left(n + 1\right)$$

In [14]:

    

S2 = Sum(1/j**2, (j, 1, oo))
S2


    

    
Out[14]:





$$\sum_{j=1}^{\infty} \frac{1}{j^{2}}$$

In [15]:

    

S2.doit()


    

    
Out[15]:





$$\frac{\pi^{2}}{6}$$

In [16]:

    

S3 = Sum(q**j, (j, 0, oo))
S3


    

    
Out[16]:





$$\sum_{j=0}^{\infty} q^{j}$$

In [17]:

    

S3.doit()


    

    
Out[17]:





$$\begin{cases} \frac{1}{- q + 1} & \text{for}\: \left|{q}\right| < 1 \\\sum_{j=0}^{\infty} q^{j} & \text{otherwise} \end{cases}$$

In [18]:

    

S3.doit().args


    

    
Out[18]:





$$\left ( \left ( \frac{1}{- q + 1}, \quad \left|{q}\right| < 1\right ), \quad \left ( \sum_{j=0}^{\infty} q^{j}, \quad \mathrm{True}\right )\right )$$

In [19]:

    

S3.doit().args[0]


    

    
Out[19]:





$$\left ( \frac{1}{- q + 1}, \quad \left|{q}\right| < 1\right )$$

In [20]:

    

S3.doit().args[0][0]


    

    
Out[20]:





$$\frac{1}{- q + 1}$$

In [21]:

    

x = Symbol('x')
y = Symbol('y')
z = x+I*y
z


    

    
Out[21]:





$$x + i y$$

In [22]:

    

w = z**4
w


    

    
Out[22]:





$$\left(x + i y\right)^{4}$$

In [23]:

    

re(w).expand()


    

    
Out[23]:





$$\left(\Re{x}\right)^{4} - 4 \left(\Re{x}\right)^{3} \Im{y} - 6 \left(\Re{x}\right)^{2} \left(\Re{y}\right)^{2} - 12 \left(\Re{x}\right)^{2} \Re{y} \Im{x} - 6 \left(\Re{x}\right)^{2} \left(\Im{x}\right)^{2} + 6 \left(\Re{x}\right)^{2} \left(\Im{y}\right)^{2} + 12 \Re{x} \left(\Re{y}\right)^{2} \Im{y} + 24 \Re{x} \Re{y} \Im{x} \Im{y} + 12 \Re{x} \left(\Im{x}\right)^{2} \Im{y} - 4 \Re{x} \left(\Im{y}\right)^{3} + \left(\Re{y}\right)^{4} + 4 \left(\Re{y}\right)^{3} \Im{x} + 6 \left(\Re{y}\right)^{2} \left(\Im{x}\right)^{2} - 6 \left(\Re{y}\right)^{2} \left(\Im{y}\right)^{2} + 4 \Re{y} \left(\Im{x}\right)^{3} - 12 \Re{y} \Im{x} \left(\Im{y}\right)^{2} + \left(\Im{x}\right)^{4} - 6 \left(\Im{x}\right)^{2} \left(\Im{y}\right)^{2} + \left(\Im{y}\right)^{4}$$

In [24]:

    

x = Symbol('x', real=True)
y = Symbol('y', real=True)
z = x+I*y
z


    

    
Out[24]:





$$x + i y$$

In [25]:

    

w = z**4
re(w)


    

    
Out[25]:





$$x^{4} - 6 x^{2} y^{2} + y^{4}$$

In [26]:

    

im(w)


    

    
Out[26]:





$$4 x^{3} y - 4 x y^{3}$$

In [27]:

    

import numpy as np


    

In [28]:

    

xn = np.linspace(0, 1, 4)
xn


    

    
Out[28]:





array([ 0.        ,  0.33333333,  0.66666667,  1.        ])

In [29]:

    

#sin(pi*xn) # mislingt
np.sin(np.pi*xn)


    

    
Out[29]:





array([  0.00000000e+00,   8.66025404e-01,   8.66025404e-01,
         1.22464680e-16])

In [30]:

    

f = exp(-x/2) * sin(pi*x)
f


    

    
Out[30]:





$$e^{- \frac{x}{2}} \sin{\left (\pi x \right )}$$

In [31]:

    

fn = lambdify(x, f, 'numpy')
fn


    

    
Out[31]:





<function numpy.<lambda>>

In [32]:

    

f.subs(x, .5), fn(.5)


    

    
Out[32]:





$$\left ( 0.778800783071405, \quad 0.778800783071\right )$$

In [33]:

    

fn(xn)


    

    
Out[33]:





array([  0.00000000e+00,   7.33074678e-01,   6.20534318e-01,
         7.42785831e-17])

In [34]:

    

plot(f, (x, -pi, pi));


    

In [35]:

    

f = sin(pi*x)
g = cos(pi*x)
p1 = plot(f, (x, -2, 2), show=False)
p2 = plot(g, (x, -2, 2), show=False)
p1.extend(p2)
p1[1].line_color = 'green'
p1.legend = True
p1.show()


    

In [36]:

    

import matplotlib.pyplot as plt


    

In [37]:

    

#%matplotlib notebook


    

In [38]:

    

%matplotlib inline


    

In [39]:

    

fn = lambdify(x, f, 'numpy')
gn = lambdify(x, g, 'numpy')
xn = np.linspace(-2, 2, 300)
plt.plot(xn, fn(xn), label='sin')
plt.plot(xn, gn(xn), label='cos')
plt.legend();


    

In [40]:

    

xn = np.linspace(-3, 3, 300)
h = exp(-x)/x
hn = lambdify(x, h, 'numpy')
plt.figure()    # schreibt sonst in den bereits geöffneten Plot,
                # wenn dieser noch nicht abgeschlossen ist
plt.plot(xn, hn(xn));


    

In [41]:

    

xp = np.linspace(.0001, 3, 150)
xm = -xp
plt.figure()
plt.plot(xp, hn(xp))
plt.plot(xm, hn(xm), 'b')  # mögliche Werte rgbcmykw
plt.axis(ymin=-10, ymax=10)
plt.title(str(h));


    

In [42]:

    

xn = np.linspace(-3, 3, 300)
plt.figure()
yn = hn(xn)
yn[abs(yn)>10] = np.nan
plt.plot(xn, yn)
plt.title(str(h));


    

In [43]:

    

xn = np.linspace(-.3, .3, 15)
yn = hn(xn)


    

    




/az076/anaconda3/envs/compana16/lib/python3.5/site-packages/numpy/__init__.py:1: RuntimeWarning: divide by zero encountered in true_divide
  """

In [44]:

    

yn


    

    
Out[44]:





array([ -4.49952936,  -5.02922724,  -5.78189079,  -6.9241629 ,
        -8.84490948, -12.71077499, -24.35507134,          inf,
        22.35445903,  10.70832512,   6.83939472,   4.91435258,
         3.76654949,   3.00711344,   2.46939407])

In [45]:

    

abs(yn) > 10


    

    
Out[45]:





array([False, False, False, False, False,  True,  True,  True,  True,
        True, False, False, False, False, False], dtype=bool)

In [46]:

    

yn[abs(yn) > 10] = np.nan
yn


    

    
Out[46]:





array([-4.49952936, -5.02922724, -5.78189079, -6.9241629 , -8.84490948,
               nan,         nan,         nan,         nan,         nan,
        6.83939472,  4.91435258,  3.76654949,  3.00711344,  2.46939407])

In [47]:

    

from sympy.plotting import plot3d


    

In [48]:

    

x = Symbol('x')
y = Symbol('y')
f = cos(sqrt(x**2+y**2))
f


    

    
Out[48]:





$$\cos{\left (\sqrt{x^{2} + y^{2}} \right )}$$

In [49]:

    

plot3d(f, (x, -3*pi, 3*pi), (y, -3*pi, 3*pi));


    

In [ ]: