o GZhg@sdZddlmZddlmZddlmZddlmZddl m Z ddl m Z ddl mZdd lmZdd lmZd&ddZd'ddZd(ddZed)ddZed)ddZed)ddZed)d d!Zd*d#d$Zd%S)+aA module for special angle formulas for trigonometric functions TODO ==== This module should be developed in the future to contain direct square root representation of .. math F(\frac{n}{m} \pi) for every - $m \in \{ 3, 5, 17, 257, 65537 \}$ - $n \in \mathbb{N}$, $0 \le n < m$ - $F \in \{\sin, \cos, \tan, \csc, \sec, \cot\}$ Without multi-step rewrites (e.g. $\tan \to \cos/\sin \to \cos/\sqrt \to \ sqrt$) or using chebyshev identities (e.g. $\cos \to \cos + \cos^2 + \cdots \to \sqrt{} + \sqrt{}^2 + \cdots $), which are trivial to implement in sympy, and had used to give overly complicated expressions. The reference can be found below, if anyone may need help implementing them. References ========== .. [*] Gottlieb, Christian. (1999). The Simple and straightforward construction of the regular 257-gon. The Mathematical Intelligencer. 21. 31-37. 10.1007/BF03024829. .. [*] https://resources.wolframcloud.com/FunctionRepository/resources/Cos2PiOverFermatPrime ) annotations)Callable)reduce)Expr)S)igcdex)Integersqrt)cacheitxintreturntuple[tuple[int, ...], int]cs|sdSt|dkrd|dfSt|dkr(t|d|d\}}|f|fSt|dd\}}t|d|\}}|gfdd|DR|fS) aNCompute extended gcd for multiple integers. Explanation =========== Given the integers $x_1, \cdots, x_n$ and an extended gcd for multiple arguments are defined as a solution $(y_1, \cdots, y_n), g$ for the diophantine equation $x_1 y_1 + \cdots + x_n y_n = g$ such that $g = \gcd(x_1, \cdots, x_n)$. Examples ======== >>> from sympy.functions.elementary._trigonometric_special import migcdex >>> migcdex() ((), 0) >>> migcdex(4) ((1,), 4) >>> migcdex(4, 6) ((-1, 1), 2) >>> migcdex(6, 10, 15) ((1, 1, -1), 1) )r)rrNc3s|]}|VqdSNr).0ivr`/var/www/auris/lib/python3.10/site-packages/sympy/functions/elementary/_trigonometric_special.py Sszmigcdex..)lenrmigcdex)r uhygrrrr.s    rdenomstuple[int, ...]cs>|sdSd dd}t||fdd |D}t|\}}|S) aCompute the partial fraction decomposition. Explanation =========== Given a rational number $\frac{1}{q_1 \cdots q_n}$ where all $q_1, \cdots, q_n$ are pairwise coprime, A partial fraction decomposition is defined as .. math:: \frac{1}{q_1 \cdots q_n} = \frac{p_1}{q_1} + \cdots + \frac{p_n}{q_n} And it can be derived from solving the following diophantine equation for the $p_1, \cdots, p_n$ .. math:: 1 = p_1 \prod_{i \ne 1}q_i + \cdots + p_n \prod_{i \ne n}q_i Where $q_1, \cdots, q_n$ being pairwise coprime implies $\gcd(\prod_{i \ne 1}q_i, \cdots, \prod_{i \ne n}q_i) = 1$, which guarantees the existence of the solution. It is sufficient to compute partial fraction decomposition only for numerator $1$ because partial fraction decomposition for any $\frac{n}{q_1 \cdots q_n}$ can be easily computed by multiplying the result by $n$ afterwards. Parameters ========== denoms : int The pairwise coprime integer denominators $q_i$ which defines the rational number $\frac{1}{q_1 \cdots q_n}$ Returns ======= tuple[int, ...] The list of numerators which semantically corresponds to $p_i$ of the partial fraction decomposition $\frac{1}{q_1 \cdots q_n} = \frac{p_1}{q_1} + \cdots + \frac{p_n}{q_n}$ Examples ======== >>> from sympy import Rational, Mul >>> from sympy.functions.elementary._trigonometric_special import ipartfrac >>> denoms = 2, 3, 5 >>> numers = ipartfrac(2, 3, 5) >>> numers (1, 7, -14) >>> Rational(1, Mul(*denoms)) 1/30 >>> out = 0 >>> for n, d in zip(numers, denoms): ... out += Rational(n, d) >>> out 1/30 rr r rrcSs||Srr)r rrrrmulszipartfrac..mulcsg|]}|qSrr)rr denomrr szipartfrac..N)r r rr rr )rr)r r"ar_rr#r ipartfracVs?   r(nlist[int] | NonecCsFg}dD]}t||\}}|dkr |}|||dkr |SqdS)z}If n can be factored in terms of Fermat primes with multiplicity of each being 1, return those primes, else None )irrN)divmodappend)r)ZprimespZquotient remainderrrr fermat_coordss r3rcCstjS)z-Computes $\cos \frac{\pi}{3}$ in square roots)rHalfrrrrcos_3sr5cCstdddS)z-Computes $\cos \frac{\pi}{5}$ in square rootsr,rr rrrrcos_5sr7c Cs|tdtddtdtdtdttddtdtddtdtdtddtdddS) z.Computes $\cos \frac{\pi}{17}$ in square rootsr- rir"r rrrrcos_17s"$ r<c)Csddd}ddd }|tjtd \}}||td \}}||td \}}||d d |d|\}} ||d d |d|\} } ||d d |d|\} } ||d d |d|\}}||d||| d| \}}|| d|| |d| \}}|| d|| | d|\}}||d||| d|\}}|| d|| | d| \}}|| d|| |d| \}}|| d|| |d|\}}||d||| d| \}}||d||||} ||d||||}!||d||||}"||d||||}#||d||||}$||d||||}%|| d|!|" }&||# d|$|% }'d||& d|'}(ttdt|(d dtjS)aComputes $\cos \frac{\pi}{257}$ in square roots References ========== .. [*] https://math.stackexchange.com/questions/516142/how-does-cos2-pi-257-look-like-in-real-radicals .. [*] https://r-knott.surrey.ac.uk/Fibonacci/simpleTrig.html r&rbrtuple[Expr, Expr]cSs0|t|d|d|t|d|dfSNrr r&r=rrrf1s0zcos_257..f1cSs|t|d|dSr?r r@rrrf2szcos_257..f2@r6r,rN)r&rr=rrr>)r&rr=rrr)rZ NegativeOnerr r4))rArBt1t2Zz1Zz3Zz2Zz4y1Zy5Zy6y2Zy3Zy7Zy8Zy4x1Zx9Zx2x10Zx3Zx11Zx4Zx12Zx5Zx13Zx6Zx14Zx15Zx7Zx8Zx16v1v2Zv3Zv4Zv5Zv6u1u2Zw1rrrcos_257s6 """""""""rRdict[int, Callable[[], Expr]]cCsttttdS)agLazily evaluated table for $\cos \frac{\pi}{n}$ in square roots for $n \in \{3, 5, 17, 257, 65537\}$. Notes ===== 65537 is the only other known Fermat prime and it is nearly impossible to build in the current SymPy due to performance issues. References ========== https://r-knott.surrey.ac.uk/Fibonacci/simpleTrig.html )r+r,r-r.)r5r7r<rRrrrr cos_tables rTN)r r rr)r r rr!)r)r rr*)rr)rrS)__doc__ __future__rtypingr functoolsrZsympy.core.exprrZsympy.core.singletonrZsympy.core.intfuncrZsympy.core.numbersrZ(sympy.functions.elementary.miscellaneousr Zsympy.core.cacher rr(r3r5r7r<rRrTrrrrs, "          ( K    *