Lektion 1

In [1]:

    

from sympy import *
init_printing()


    

In [2]:

    

2+2


    

    
Out[2]:





$$4$$

In [3]:

    

2*3


    

    
Out[3]:





$$6$$

In [4]:

    

2**3


    

    
Out[4]:





$$8$$

In [5]:

    

1/3


    

    
Out[5]:





$$0.3333333333333333$$

In [6]:

    

1/0


    

    




---------------------------------------------------------------------------
ZeroDivisionError                         Traceback (most recent call last)
<ipython-input-6-05c9758a9c21> in <module>()
----> 1 1/0

ZeroDivisionError: division by zero

In [7]:

    

3 * (1/3)


    

    
Out[7]:





$$1.0$$

In [8]:

    

3**100 * (1/3)**100


    

    
Out[8]:





$$0.9999999999999944$$

In [9]:

    

drittel = Rational(1,3)
drittel


    

    
Out[9]:





$$\frac{1}{3}$$

In [10]:

    

3**100 * drittel**100


    

    
Out[10]:





$$1$$

In [11]:

    

drittel**100


    

    
Out[11]:





$$\frac{1}{515377520732011331036461129765621272702107522001}$$

In [12]:

    

3**100


    

    
Out[12]:





$$515377520732011331036461129765621272702107522001$$

In [13]:

    

(1/3)**1000


    

    
Out[13]:





$$0.0$$

In [14]:

    

3**1000 * (1/3)**1000


    

    




---------------------------------------------------------------------------
OverflowError                             Traceback (most recent call last)
<ipython-input-14-4c3ad28e3b02> in <module>()
----> 1 3**1000 * (1/3)**1000

OverflowError: int too large to convert to float

In [15]:

    

pi


    

    
Out[15]:





$$\pi$$

In [16]:

    

N(pi, 20)


    

    
Out[16]:





$$3.1415926535897932385$$

In [17]:

    

N(pi, 200)


    

    
Out[17]:





$$3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930382$$

In [18]:

    

print(N(pi,200))


    

    




3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303820

In [19]:

    

N(1/3, 200)


    

    
Out[19]:





$$0.333333333333333314829616256247390992939472198486328125$$

In [20]:

    

N(Rational(1,3), 200)


    

    
Out[20]:





$$0.33333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333$$

In [21]:

    

N(Rational(1,3), 200)**1000


    

    
Out[21]:





$$7.5638913231040998047574498974057316364042448856095234455200663316710977361426404768891548707455227125428677184833301291900453415960602024847503910604919173382646766637018179101786118311000891082040681 \cdot 10^{-478}$$

In [22]:

    

N(Rational(1,3), 200)**1000 * 3**1000


    

    
Out[22]:





$$1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004$$

In [23]:

    

3


    

    
Out[23]:





$$3$$

In [24]:

    

S(3)


    

    
Out[24]:





$$3$$

In [25]:

    

type(3)


    

    
Out[25]:





int

In [26]:

    

type(S(3))


    

    
Out[26]:





sympy.core.numbers.Integer

In [27]:

    

S(1)/3


    

    
Out[27]:





$$\frac{1}{3}$$

In [28]:

    

S(0.1)


    

    
Out[28]:





$$0.1$$

In [29]:

    

x = S('x')
x


    

    
Out[29]:





$$x$$

In [30]:

    

type(x)


    

    
Out[30]:





sympy.core.symbol.Symbol

In [31]:

    

type(drittel)


    

    
Out[31]:





sympy.core.numbers.Rational

In [32]:

    

y = S('y')
y


    

    
Out[32]:





$$y$$

In [33]:

    

f = (x - y)**2
f


    

    
Out[33]:





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

In [34]:

    

x = 5


    

In [35]:

    

f


    

    
Out[35]:





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

In [36]:

    

x


    

    
Out[36]:





$$5$$

In [37]:

    

f.atoms()


    

    
Out[37]:





$$\left\{-1, 2, x, y\right\}$$

In [38]:

    

sqrt(81)


    

    
Out[38]:





$$9$$

In [39]:

    

sqrt(-81)


    

    
Out[39]:





$$9 i$$

In [40]:

    

I**2


    

    
Out[40]:





$$-1$$

In [41]:

    

sqrt(243)


    

    
Out[41]:





$$9 \sqrt{3}$$

In [42]:

    

sqrt(243.)


    

    
Out[42]:





$$15.5884572681199$$

In [43]:

    

sqrt(9*y**2)


    

    
Out[43]:





$$3 \sqrt{y^{2}}$$

In [44]:

    

factorial(5)


    

    
Out[44]:





$$120$$

In [45]:

    

factorial(70)


    

    
Out[45]:





$$11978571669969891796072783721689098736458938142546425857555362864628009582789845319680000000000000000$$

In [46]:

    

N(factorial(70))


    

    
Out[46]:





$$1.19785716699699 \cdot 10^{100}$$

In [47]:

    

sin(pi)


    

    
Out[47]:





$$0$$

In [48]:

    

cos(pi)


    

    
Out[48]:





$$-1$$

In [49]:

    

tan(pi/2)


    

    
Out[49]:





$$\tilde{\infty}$$

In [50]:

    

print(tan(pi/2))


    

    




zoo

In [51]:

    

?zoo


    

In [52]:

    

alpha = Symbol('alpha')
alpha


    

    
Out[52]:





$$\alpha$$

In [53]:

    

exp(1)


    

    
Out[53]:





$$e$$

In [54]:

    

e


    

    




---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-54-9ffbf43126e3> in <module>()
----> 1 e

NameError: name 'e' is not defined

In [55]:

    

log(exp(1))


    

    
Out[55]:





$$1$$

In [56]:

    

abs(-1)


    

    
Out[56]:





$$1$$

In [57]:

    

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


    

In [58]:

    

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


    

    
Out[58]:





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

In [59]:

    

f.expand()


    

    
Out[59]:





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

In [60]:

    

expand(f)


    

    
Out[60]:





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

In [61]:

    

f.expand().factor()


    

    
Out[61]:





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

In [62]:

    

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


    

    
Out[62]:





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

In [63]:

    

g.ratsimp()


    

    
Out[63]:





$$x + y$$

In [64]:

    

g.simplify()


    

    
Out[64]:





$$x + y$$

In [65]:

    

h = x*x**y
h


    

    
Out[65]:





$$x x^{y}$$

In [66]:

    

h.powsimp()


    

    
Out[66]:





$$x^{y + 1}$$

In [67]:

    

h.powsimp().expand()


    

    
Out[67]:





$$x x^{y}$$

In [68]:

    

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


    

    
Out[68]:





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

In [69]:

    

f.simplify()


    

    
Out[69]:





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

In [70]:

    

f.ratsimp()


    

    
Out[70]:





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

In [71]:

    

f.trigsimp()


    

    
Out[71]:





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

In [72]:

    

f.expand()


    

    
Out[72]:





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

In [73]:

    

f.expand(numer=True)


    

    
Out[73]:





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

In [74]:

    

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


    

    
Out[74]:





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

In [75]:

    

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


    

    
Out[75]:





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

In [76]:

    

f.expand(gibtsnich=True)


    

    
Out[76]:





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

In [ ]: