Lektion 6



In [1]:

    
from sympy import *
init_printing()
import numpy as np
import matplotlib.pyplot as plt

interaktiv



In [2]:

    
#%matplotlib notebook

für den Druck



In [3]:

    
%matplotlib inline

Schleifen

Blöcke werden durch Einrückung festgelegt



In [4]:

    
summe = 0
for j in range(10):
    summe = summe + j
summe









    Out[4]:





$$45$$

Probe



In [5]:

    
np.arange(10).sum()









    Out[5]:





45

if/elif/else



In [6]:

    
12 % 3  # modulo









    Out[6]:





$$0$$



In [7]:

    
for a in range(7):
    if a % 3 == 0:
        print(a, 'ist durch 3 teilbar')
    elif a % 3 == 1:
        print(a, 'lässt Rest 1')
    else:
        print(a, 'lässt Rest 2')









    



0 ist durch 3 teilbar
1 lässt Rest 1
2 lässt Rest 2
3 ist durch 3 teilbar
4 lässt Rest 1
5 lässt Rest 2
6 ist durch 3 teilbar

Pascalsches Dreieck



In [8]:

    
N = 6



In [9]:

    
b = {}
for n in range(1, N+1):
    b[(n,1)] = 1
    for j in range(2,n):
        b[(n,j)] = b[(n-1,j-1)] + b[(n-1,j)]
    b[(n,n)] = 1



In [10]:

    
for n in range(1,N+1):
    print([b[(n,j)] for j in range(1,n+1)])









    



[1]
[1, 1]
[1, 2, 1]
[1, 3, 3, 1]
[1, 4, 6, 4, 1]
[1, 5, 10, 10, 5, 1]



In [11]:

    
for j in range(1, 6):
    wert = sin(2*pi/j)
    print(wert)









    



0
0
sqrt(3)/2
1
sqrt(sqrt(5)/8 + 5/8)



In [12]:

    
from IPython.display import display, Image



In [13]:

    
for j in range(1, 11):
    wert = sin(2*pi/j)
    display((2*pi/j, wert))









    





$$\left ( 2 \pi, \quad 0\right )$$






    





$$\left ( \pi, \quad 0\right )$$






    





$$\left ( \frac{2 \pi}{3}, \quad \frac{\sqrt{3}}{2}\right )$$






    





$$\left ( \frac{\pi}{2}, \quad 1\right )$$






    





$$\left ( \frac{2 \pi}{5}, \quad \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right )$$






    





$$\left ( \frac{\pi}{3}, \quad \frac{\sqrt{3}}{2}\right )$$






    





$$\left ( \frac{2 \pi}{7}, \quad \sin{\left (\frac{2 \pi}{7} \right )}\right )$$






    





$$\left ( \frac{\pi}{4}, \quad \frac{\sqrt{2}}{2}\right )$$






    





$$\left ( \frac{2 \pi}{9}, \quad \sin{\left (\frac{2 \pi}{9} \right )}\right )$$






    





$$\left ( \frac{\pi}{5}, \quad \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}\right )$$

Mehrere Zeichnungen in einem Bild



In [14]:

    
x = Symbol('x')
y = Symbol('y')



In [15]:

    
h = (x**2+y**2)**2 + 3*x**2*y - y**3
xn = np.linspace(-1, 1, 200)
yn = xn
hn = lambdify((x,y), h)
X, Y = np.meshgrid(xn, yn)



In [16]:

    
fig = plt.figure()
for j in range(9):
    fig.add_subplot(331+j)
    plt.contour(X, Y, hn(X, Y), [.001*(j-4)])
    plt.axis('equal')
    if j % 3 != 0:
        plt.yticks([])
    if j <= 5:
        plt.xticks([])
    else:
        plt.xticks([-1.0, 0.0, 1.0])
fig.savefig('9plots.png')  # pdf geht auch



In [17]:

    
Image('9plots.png')  # kann pdf nicht anzeigen









    Out[17]:

Funktionen



In [18]:

    
x = Symbol('x')
xn = np.arange(6)
zn = 1j + xn
x, xn, zn









    Out[18]:





(x,
 array([0, 1, 2, 3, 4, 5]),
 array([ 0.+1.j,  1.+1.j,  2.+1.j,  3.+1.j,  4.+1.j,  5.+1.j]))



In [19]:

    
def f(x):
    return x**3 + 2*x + 15



In [20]:

    
f(0)









    Out[20]:





$$15$$



In [21]:

    
f(0.)









    Out[21]:





$$15.0$$



In [22]:

    
f(0+0j)









    Out[22]:





(15+0j)



In [23]:

    
f(2*x)









    Out[23]:





$$8 x^{3} + 4 x + 15$$



In [24]:

    
f(xn)









    Out[24]:





array([ 15,  18,  27,  48,  87, 150])

Duck typing:

“When I see a bird that walks like a duck and swims like a duck and quacks like a duck, I call that bird a duck.”

– James Whitcomb Riley



In [25]:

    
def g(x):
    return sin(x) + f(x)



In [26]:

    
g(-2*x)









    Out[26]:





$$- 8 x^{3} - 4 x - \sin{\left (2 x \right )} + 15$$



In [27]:

    
g(pi)









    Out[27]:





$$2 \pi + 15 + \pi^{3}$$



In [28]:

    
g(3.1415)









    Out[28]:





$$52.2866260519647$$



In [29]:

    
# g(xn)  zeigt einen AttributeError



In [30]:

    
def abss(x):
    if x > 0:
        return x
    else:
        return -x



In [31]:

    
abss(-5)









    Out[31]:





$$5$$



In [32]:

    
# abss(x)  # TypeError



In [33]:

    
xp = Symbol('x_+', positive=True)
abss(xp)









    Out[33]:





$$x_{+}$$

Zusammenfassen



In [34]:

    
x = Symbol('x')



In [35]:

    
a = (x**4-3*x**2+5)*exp(x**2)*sin(x)
a









    Out[35]:





$$\left(x^{4} - 3 x^{2} + 5\right) e^{x^{2}} \sin{\left (x \right )}$$



In [36]:

    
da = diff(a, x, 3)
da









    Out[36]:





$$\left(24 x^{3} \left(2 x^{2} - 3\right) \sin{\left (x \right )} + 8 x^{3} \left(x^{4} - 3 x^{2} + 5\right) \sin{\left (x \right )} + 24 x^{2} \left(2 x^{2} - 3\right) \cos{\left (x \right )} + 12 x^{2} \left(x^{4} - 3 x^{2} + 5\right) \cos{\left (x \right )} + 6 x \left(2 x^{2} - 3\right) \sin{\left (x \right )} + 36 x \left(2 x^{2} - 1\right) \sin{\left (x \right )} + 6 x \left(x^{4} - 3 x^{2} + 5\right) \sin{\left (x \right )} + 24 x \sin{\left (x \right )} + 18 \left(2 x^{2} - 1\right) \cos{\left (x \right )} + 5 \left(x^{4} - 3 x^{2} + 5\right) \cos{\left (x \right )}\right) e^{x^{2}}$$



In [37]:

    
da.expand()









    Out[37]:





$$8 x^{7} e^{x^{2}} \sin{\left (x \right )} + 12 x^{6} e^{x^{2}} \cos{\left (x \right )} + 30 x^{5} e^{x^{2}} \sin{\left (x \right )} + 17 x^{4} e^{x^{2}} \cos{\left (x \right )} + 34 x^{3} e^{x^{2}} \sin{\left (x \right )} + 9 x^{2} e^{x^{2}} \cos{\left (x \right )} + 7 e^{x^{2}} \cos{\left (x \right )}$$



In [38]:

    
da.expand().collect(exp(x**2))









    Out[38]:





$$\left(8 x^{7} \sin{\left (x \right )} + 12 x^{6} \cos{\left (x \right )} + 30 x^{5} \sin{\left (x \right )} + 17 x^{4} \cos{\left (x \right )} + 34 x^{3} \sin{\left (x \right )} + 9 x^{2} \cos{\left (x \right )} + 7 \cos{\left (x \right )}\right) e^{x^{2}}$$



In [39]:

    
rcollect(collect(da.expand(), exp(x**2)), cos(x), sin(x))









    Out[39]:





$$\left(\left(8 x^{7} + 30 x^{5} + 34 x^{3}\right) \sin{\left (x \right )} + \left(12 x^{6} + 17 x^{4} + 9 x^{2} + 7\right) \cos{\left (x \right )}\right) e^{x^{2}}$$

Trigonometrische Funktionen



In [40]:

    
x = Symbol('x')
y = Symbol('y')



In [41]:

    
a = cos(x+y)



In [42]:

    
A = expand(a, trig=True)
A









    Out[42]:





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



In [43]:

    
trigsimp(A)









    Out[43]:





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



In [44]:

    
c = tan(x)**2 + 1
c









    Out[44]:





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



In [45]:

    
c.trigsimp()









    Out[45]:





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



In [46]:

    
d = tan(3*x)
d.simplify()









    Out[46]:





$$\tan{\left (3 x \right )}$$



In [47]:

    
d.expand(trig=True)









    Out[47]:





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



In [48]:

    
d.expand(trig=True).ratsimp()









    Out[48]:





$$\frac{1}{3} \tan{\left (x \right )} - \frac{8 \tan{\left (x \right )}}{9 \tan^{2}{\left (x \right )} - 3}$$

benutzerspezifische trigonometrische Vereinfachungen



In [49]:

    
b = sin(x) + sin(y)
b









    Out[49]:





$$\sin{\left (x \right )} + \sin{\left (y \right )}$$



In [50]:

    
b.trigsimp(method='fu')









    Out[50]:





$$\sin{\left (x \right )} + \sin{\left (y \right )}$$

Wie kompliziert ist dieser Ausdruck?



In [51]:

    
b.count_ops(visual=True)









    Out[51]:





$$ADD + 2 SIN$$



In [52]:

    
def my_measure(expr):
    opc = expr.count_ops(visual=True)
#    print(opc)   # zur Fehlersuche
    wert = {}
    wert['ADD'] = 1
    wert['SIN'] = 1
    wert['MUL'] = -100
    return opc.subs(wert)



In [53]:

    
my_measure(b)









    Out[53]:





$$3$$



In [54]:

    
# b.trigsimp(method='fu', measure=my_measure) # TypeError



In [55]:

    
def my_measure(expr):
    opc = expr.count_ops(visual=True)
    wert = {}
    wert['ADD'] = 1
    wert['SIN'] = 1
    wert['MUL'] = -100
    wert['COS'] = 1
    wert['DIV'] = 1 
    wert['SUB'] = 1
    return opc.subs(wert)



In [56]:

    
b.trigsimp(method='fu', measure=my_measure)









    Out[56]:





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



In [57]:

    
e = tan(x) + cot(y)
e.simplify()









    Out[57]:





$$\tan{\left (x \right )} + \cot{\left (y \right )}$$



In [58]:

    
def my_measure(expr):
    opc = expr.count_ops(visual=True)
#    print(opc)   # zur Fehlersuche
    wert = {}
    wert['ADD'] = 1
    wert['SIN'] = 1
    wert['COS'] = 1
    wert['TAN'] = 100 
    wert['COT'] = 100
    return opc.subs(wert)



In [59]:

    
# e.trigsimp(method='fu', measure=my_measure) # TypeError



In [60]:

    
def my_measure(expr):
    opc = expr.count_ops(visual=True)
    wert = {}
    wert['ADD'] = 1
    wert['SIN'] = 1
    wert['COS'] = 1
    wert['TAN'] = 100
    wert['COT'] = 100
    wert['DIV'] = 1
    return opc.subs(wert)



In [61]:

    
e1 = e.trigsimp(method='fu', measure=my_measure)
e1









    Out[61]:





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



In [62]:

    
e2 = e1.ratsimp()
e2









    Out[62]:





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



In [63]:

    
e2.trigsimp()









    Out[63]:





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



In [64]:

    
num, den = numer(e2), denom(e2)
num1 = num.trigsimp()
num1/den









    Out[64]:





$$\frac{\cos{\left (x - y \right )}}{\sin{\left (y \right )} \cos{\left (x \right )}}$$

Potenzfunktionen



In [65]:

    
f = x**Rational(1,3)



In [66]:

    
plot(f, (x, -1, 1));



In [67]:

    
z = f.subs(x, -1)
z









    Out[67]:





$$\sqrt[3]{-1}$$



In [68]:

    
expand(z, complex=True)









    Out[68]:





$$\frac{1}{2} + \frac{\sqrt{3} i}{2}$$



In [69]:

    
expand(exp(pi*I/3), complex=True)









    Out[69]:





$$\frac{1}{2} + \frac{\sqrt{3} i}{2}$$



In [70]:

    
g = sign(x)*abs(x)**Rational(1,3)
g









    Out[70]:





$$\sqrt[3]{\left|{x}\right|} \operatorname{sign}{\left (x \right )}$$



In [71]:

    
plot(g, (x,-1,1));