CoCalc Public Filesscratch / formula.md
Authors: Harald Schilly, ℏal Snyder
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test

$\displaystyle {{\varepsilon}}^{3} f_{1}\left(x_{0}, y, t + t_{2}\right) f_{1}\left(x_{0}, y, t\right) \mathrm{D}_{0, 0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}}^{3} f_{1}\left(x_{0}, y, t\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{2}\right) + {{\varepsilon}}^{2} f_{1}\left(x_{0}, y, t + t_{2}\right) f_{1}\left(x_{0}, y, t\right) \mathrm{D}_{0, 0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}}^{2} f_{1}\left(x_{0}, y, t\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{2}\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}}^{2} f_{0}\left(x_{0}, y, t\right) f_{1}\left(x_{0}, y, t + t_{2}\right) \mathrm{D}_{0, 0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}}^{2} f_{0}\left(x_{0}, y, t + t_{2}\right) f_{1}\left(x_{0}, y, t\right) \mathrm{D}_{0, 0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}}^{2} f_{1}\left(x_{0}, y, t\right) g_{1}\left(y, t + t_{2}\right) \mathrm{D}_{0, 1}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}}^{2} f_{1}\left(x_{0}, y, t + t_{2}\right) g_{1}\left(y, t\right) \mathrm{D}_{0, 1}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}}^{2} f_{1}\left(x_{0}, y, t\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{2}\right) + {{\varepsilon}}^{2} f_{0}\left(x_{0}, y, t\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{2}\right) + {{\varepsilon}}^{2} g_{1}\left(y, t\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{1}\left(f_{1}\right)\left(x_{0}, y, t + t_{2}\right) + {{\varepsilon}} f_{0}\left(x_{0}, y, t\right) f_{1}\left(x_{0}, y, t + t_{2}\right) \mathrm{D}_{0, 0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} f_{0}\left(x_{0}, y, t + t_{2}\right) f_{1}\left(x_{0}, y, t\right) \mathrm{D}_{0, 0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} f_{1}\left(x_{0}, y, t\right) g_{1}\left(y, t + t_{2}\right) \mathrm{D}_{0, 1}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} f_{1}\left(x_{0}, y, t + t_{2}\right) g_{1}\left(y, t\right) \mathrm{D}_{0, 1}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} f_{1}\left(x_{0}, y, t\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{2}\right) + {{\varepsilon}} f_{0}\left(x_{0}, y, t\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{2}\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} g_{1}\left(y, t\right) \mathrm{D}_{1}\left(f_{0}\right)\left(x_{0}, y, t + t_{2}\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} f_{0}\left(x_{0}, y, t + t_{2}\right) f_{0}\left(x_{0}, y, t\right) \mathrm{D}_{0, 0}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} f_{0}\left(x_{0}, y, t\right) g_{1}\left(y, t + t_{2}\right) \mathrm{D}_{0, 1}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} f_{0}\left(x_{0}, y, t + t_{2}\right) g_{1}\left(y, t\right) \mathrm{D}_{0, 1}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} g_{1}\left(y, t + t_{2}\right) g_{1}\left(y, t\right) \mathrm{D}_{1, 1}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + {{\varepsilon}} f_{0}\left(x_{0}, y, t\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{0}\left(f_{1}\right)\left(x_{0}, y, t + t_{2}\right) + {{\varepsilon}} g_{1}\left(y, t\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{1}\left(f_{1}\right)\left(x_{0}, y, t + t_{2}\right) + {{\varepsilon}} g_{1}\left(y, t\right) \mathrm{D}_{1}\left(f_{1}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{0}\left(g_{1}\right)\left(y, t + t_{2}\right) + f_{0}\left(x_{0}, y, t + t_{2}\right) f_{0}\left(x_{0}, y, t\right) \mathrm{D}_{0, 0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + f_{0}\left(x_{0}, y, t\right) g_{1}\left(y, t + t_{2}\right) \mathrm{D}_{0, 1}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + f_{0}\left(x_{0}, y, t + t_{2}\right) g_{1}\left(y, t\right) \mathrm{D}_{0, 1}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + g_{1}\left(y, t + t_{2}\right) g_{1}\left(y, t\right) \mathrm{D}_{1, 1}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) + f_{0}\left(x_{0}, y, t\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{2}\right) + g_{1}\left(y, t\right) \mathrm{D}_{0}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{1}\left(f_{0}\right)\left(x_{0}, y, t + t_{2}\right) + g_{1}\left(y, t\right) \mathrm{D}_{1}\left(f_{0}\right)\left(x_{0}, y, t + t_{1} + t_{2}\right) \mathrm{D}_{0}\left(g_{1}\right)\left(y, t + t_{2}\right)$

An elliptic curve is (in its simplest form) an equation of the form $y^2 = x^3 + Ax + B$ for some values $A$ and $B$. (Note that the plot of such a curve is not an ellipse! The name arises from the original study of these curves in connection with the computation of the arclength of a sector of an ellipse.)

There is an $n_0 > 0$ such that $a_n < 2^n$ for all $n > n_0$.

broken $\int_0^\infty x^5\, \mathrm{d}x$ ?

abc text $123 abc single $abc line

abc $a^b$ test

$a^b$

asdf $a^b$ asdf
asdf

$cd ~$ mkdir apps

$nosetests pandas$
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