Python - `SymPy`, and $\LaTeX$



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%matplotlib inline

import sympy as sp
import numpy as np
import matplotlib.pyplot as plt

Symbolic Mathematics (`SymPy`)



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sp.init_printing()     # Turns on pretty printing



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np.sqrt(8)



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sp.sqrt(8)

You have to explicitly tell `SymPy` what symbols you want to use.



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x, y, z  = sp.symbols('x y z')

Expressions are then able use these symbols



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my_equation = 2 * x + y
my_equation



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my_equation + 3



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my_equation - x



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my_equation / x

`SymPy` has all sorts of ways to manipulates symbolic equations



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sp.simplify(my_equation / x)



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another_equation = (x + 2) * (x - 3)
another_equation



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sp.expand(another_equation)



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long_equation = 2*y*x**3 + 12*x**2 - x + 3 - 8*x**2 + 4*x + x**3 + 5 + 2*y*x**2 + x*y
long_equation



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sp.collect(long_equation,x)



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sp.collect(long_equation,y)

And solve equations - `solve`



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yet_another_equation = 2 * x**2 - 5 * x + 30

yet_another_equation



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sp.solve(yet_another_equation,x)

Calculus



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yet_another_equation



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sp.diff(yet_another_equation,x)



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sp.diff(yet_another_equation,x,2)



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sp.integrate(yet_another_equation,x)



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sp.integrate(yet_another_equation,(x,0,5))   # limits x = 0 to 5

Taylor Expansions



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still_another_equation = sp.sin(x) * sp.exp(-x)
still_another_equation



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sp.series(still_another_equation, x)



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sp.series(still_another_equation, x, x0 = 0, n = 8)



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sp.series(still_another_equation, x, x0 = sp.pi, n = 8)

Limits

$$\lim _{x\to \infty }\left(1+{\frac {1}{x}}\right)^{x}$$



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limit_equation = (1 + (1 / x)) ** x

limit_equation



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sp.limit(limit_equation, x, sp.oo)      # sp.oo = infinity

System of equations

$$ \begin{array}{c} x + 3y + 5z = 10 \\ 2x + 5y + z = 8 \\ 2x + 3y + 8z = 3 \\ \end{array} \hspace{3cm} \left[ \begin{array}{ccc} 1 & 3 & 5 \\ 2 & 5 & 1 \\ 2 & 3 & 8 \end{array} \right] \left[ \begin{array}{c} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{c} 10\\ 8\\ 3 \end{array} \right] $$



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AA = sp.Matrix([[1,3,5],[2,5,1],[2,3,8]])
bb = sp.Matrix([[10],[8],[3]])



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AA, bb



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AA**-1



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AA**-1 * AA



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AA**-1 * bb

General Equation Solving - `nsolve`

$$ \large y_1 = 10\,\sin(5x) \ e^{-x}\\ \large y_2 = 6 - e^{0.75x} $$

Where do they cross? - The graph



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my_x = np.linspace(0,2*np.pi,100)



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my_y1 = 10 * np.sin(5*my_x) * np.exp(-my_x)
my_y2 = 6 - np.exp(0.75 * my_x)



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fig,ax = plt.subplots(1,1)
fig.set_size_inches(10,4)

fig.tight_layout()

ax.set_ylim(-5,8)
ax.set_xlim(0,3)

ax.set_xlabel("This is X")
ax.set_ylabel("This is Y")

ax.plot(my_x, my_y1, color='b', marker='None', linestyle='--')
ax.plot(my_x, my_y2, color='r', marker='None', linestyle='-');

Where do they cross? - The `sympy` solution



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equation_one = 10 * sp.sin(5*x) * sp.exp(-x)
equation_two = 6 - sp.exp(0.75 * x)



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equation_one, equation_two



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my_guess = 0.5

sp.nsolve(equation_one - equation_two, x, my_guess)



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all_guesses = (0.1, 0.5, 2.5)

for i in all_guesses:
    result = sp.nsolve(equation_one - equation_two, x, i)
    print(result)

Your guess has to be (somewhat) close or the solution will not converge:



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my_guess = 200

sp.nsolve(equation_one - equation_two, x, my_guess)

Solve ODE Equations - `dsolve`



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f = sp.Function('f')



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equation_ode = sp.Derivative(f(x), x, x) + 9*f(x)

equation_ode



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sp.dsolve(equation_ode, f(x))

`SymPy` can do so much more. It really is magic.

Complete documentation can be found here

$\LaTeX$

Python uses the $\LaTeX$ language to typeset equations.

Most LaTeX commands are prefixed with a "\". For example \pi is the command to produce the lower case Greek letter pi. The characters # $ % & ~ _ ^ \ { } are special characters in LaTeX. If you want to typeset them you need to put a \ in front of them. For example \$ will typeset the symbol $ % - comment. Everything is ignored after a % $ - Math mode. Start and Stop math mode. $\pi$ ^ - Superscript in Math mode. $2^2$ _ - Subscript in Math mode. $2_2$

## Latex Symbols
## Latex draw symbols

Use a single set of `$` to make your $\LaTeX$ inline and a double set `$$` to center

$$ \int \cos(x)\ dx = \sin(x) $$

This code will produce the output:

$$ \int \cos(x)\ dx = \sin(x) $$

Use can use $\LaTeX$ in plots:



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plt.style.use('ggplot')

my_a = np.linspace(0,2*np.pi,100)
my_b = np.sin(5*my_a) * np.exp(-my_a)



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fig,ax = plt.subplots(1,1)
fig.set_size_inches(10,4)

fig.tight_layout()

ax.plot(my_a, my_b, color='r', marker='None', linestyle='-');

ax.set_title("A gratuitous use of $σύμβολον$")
ax.set_xlabel("This is in units of 2$\pi$")
ax.set_ylabel("This is B")

ax.text(2.0, 0.4, '$y\ =\ \sin(5a)\ e^{-a}$', color='green', fontsize=36);

Use can use `SymPy` to make $\LaTeX$ equations for you!



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a =  1/( ( z + 2 ) * ( z + 1 ) )

print(sp.latex(a))

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



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print(sp.latex(sp.Integral(z**2,z)))

$$ \int z^{2}\, dz $$

`Astropy` can output $\LaTeX$ tables



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from astropy.io import ascii
from astropy.table import QTable



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my_table = QTable.read('./MyData/Zodiac.csv', format='ascii.csv')



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my_table[0:3]



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ascii.write(my_table, format='latex')

Python - SymPy, and $\LaTeX$

Symbolic Mathematics (SymPy)

You have to explicitly tell SymPy what symbols you want to use.

Expressions are then able use these symbols

SymPy has all sorts of ways to manipulates symbolic equations

And solve equations - solve

Calculus

Taylor Expansions

Limits

System of equations

General Equation Solving - nsolve

Where do they cross? - The graph

Where do they cross? - The sympy solution

Your guess has to be (somewhat) close or the solution will not converge:

Solve ODE Equations - dsolve

SymPy can do so much more. It really is magic.

Complete documentation can be found here

$\LaTeX$

Python uses the $\LaTeX$ language to typeset equations.

Use a single set of $ to make your $\LaTeX$ inline and a double set $$ to center

This code will produce the output:

$$ \int \cos(x)\ dx = \sin(x) $$

Use can use $\LaTeX$ in plots:

Use can use SymPy to make $\LaTeX$ equations for you!

Astropy can output $\LaTeX$ tables

Python - `SymPy`, and $\LaTeX$

Symbolic Mathematics (`SymPy`)

You have to explicitly tell `SymPy` what symbols you want to use.

`SymPy` has all sorts of ways to manipulates symbolic equations

And solve equations - `solve`

General Equation Solving - `nsolve`

Where do they cross? - The `sympy` solution

Solve ODE Equations - `dsolve`

`SymPy` can do so much more. It really is magic.

Use a single set of `$` to make your $\LaTeX$ inline and a double set `$$` to center

Use can use `SymPy` to make $\LaTeX$ equations for you!

`Astropy` can output $\LaTeX$ tables