U —9%eMã@sòdZddlmZddlmZddlmZddlmZddl m Z ddl m Z m Z ddlmZdd lmZd d d œd d„Zd ddœdd„Zd ddœdd„Zeddœdd„ƒZeddœdd„ƒZeddœdd„ƒZeddœdd „ƒZd!dœd"d#„Zd$S)%aÛA module for special angle forumlas for trigonometric functions TODO ==== This module should be developed in the future to contain direct squrae 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 é)Ú annotations)ÚCallable)Úreduce)ÚExpr)ÚS)ÚigcdexÚInteger©Úsqrt)ÚcacheitÚintztuple[tuple[int, ...], int])ÚxÚreturncs”|sdSt|ƒdkr d|dfSt|ƒdkrPt|d|dƒ\}‰}|ˆf|fSt|dd…Ž\}}t|d|ƒ\}‰}|f‡fdd„|Dƒ˜|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) )©ré)rréNc3s|]}ˆ|VqdS©Nr)Ú.0Úi©Úvrúp/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/sympy/functions/elementary/_trigonometric_special.pyÚ Rszmigcdex..)ÚlenrÚmigcdex)r ÚuÚhÚyÚgrrrr-s    rztuple[int, ...])ÚdenomsrcsF|sdSddddœdd„}t||ƒ‰‡fdd„|Dƒ}t|Ž\}}|S)aâCompute the 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 existance 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 rrrrÚmul—szipartfrac..mulcsg|] }ˆ|‘qSrr)rr ©ÚdenomrrÚ ›szipartfrac..)rr)rr ÚarÚ_rr!rÚ ipartfracUs?  r&zlist[int] | None)ÚnrcCsFg}dD]8}t||ƒ\}}|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 )ééééirrN)ÚdivmodÚappend)r'ZprimesÚpZquotientÚ remainderrrrÚ fermat_coords s  r0r)rcCstjS)z-Computes $\cos \frac{\pi}{3}$ in square roots)rÚHalfrrrrÚcos_3°sr2cCstdƒddS)z-Computes $\cos \frac{\pi}{5}$ in square rootsr)rér rrrrÚcos_5¶sr4c Cs|tdtdƒdtdƒtdtdƒƒttdƒdtdtdƒƒdtdƒtdtdƒƒdtdƒdƒdƒS) z.Computes $\cos \frac{\pi}{17}$ in square rootsér*é riøÿÿÿréé"r rrrrÚcos_17¼s"$ÿ ÿÿÿþÿr9c)Csâddddœdd„}ddddœdd„}|tjtdƒƒ\}}||td ƒƒ\}}||td ƒƒ\}}||d d |d |ƒ\}} ||d d |d |ƒ\} } ||d d |d |ƒ\} } ||d d |d |ƒ\}}||d ||| d | ƒ\}}|| d || |d | ƒ\}}|| d || | d |ƒ\}}||d ||| d |ƒ\}}|| d || | d | ƒ\}}|| d || |d | ƒ\}}|| d || |d |ƒ\}}||d ||| d | ƒ\}}||d ||||ƒ} ||d ||||ƒ}!||d ||||ƒ}"||d ||||ƒ}#||d ||||ƒ}$||d ||||ƒ}%|| d |!|"ƒ }&||# d |$|%ƒ }'d||& d |'ƒ}(ttd ƒt|(d ƒdtjƒS)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 rztuple[Expr, Expr])r$ÚbrcSs0|t|d|ƒd|t|d|ƒdfS©Nrr ©r$r:rrrÚf1Ïszcos_257..f1cSs|t|d|ƒdSr;r r<rrrÚf2Òszcos_257..f2éé@r3r)réüÿÿÿéþÿÿÿé)rZ NegativeOnerr r1))r=r>Út1Út2Zz1Zz3Zz2Zz4Úy1Zy5Zy6Úy2Zy3Zy7Zy8Zy4Úx1Zx9Zx2Úx10Zx3Zx11Zx4Zx12Zx5Zx13Zx6Zx14Zx15Zx7Zx8Zx16Zv1Zv2Zv3Zv4Zv5Zv6Úu1Úu2Zw1rrrÚcos_257Ås6 """"""""rLzdict[int, Callable[[], Expr]]cCsttttdœS)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+)r2r4r9rLrrrrÚ cos_tableðs ürMN)Ú__doc__Ú __future__rÚtypingrÚ functoolsrZsympy.core.exprrZsympy.core.singletonrZsympy.core.numbersrrZ(sympy.functions.elementary.miscellaneousr Zsympy.core.cacher rr&r0r2r4r9rLrMrrrrÚs("       (K*