U L?hg@sdZddlmZddlmZddlmZddlmZddl m Z ddl m Z ddl mZdd lmZdd lmZd d d ddZd ddddZd ddddZeddddZeddddZeddddZeddd d!Zd"dd#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)cacheitintztuple[tuple[int, ...], int])xreturncs|sdSt|dkr d|dfSt|dkrPt|d|d\}}|f|fSt|dd\}}t|d|\}}|ffdd|D|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).0ivrc/var/www/html/venv/lib/python3.8/site-packages/sympy/functions/elementary/_trigonometric_special.py Sszmigcdex..)lenrmigcdex)r uhygrrrr.s    rztuple[int, ...])denomsrcsF|sdSdddddd}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..)rr)rr ar_rr!r ipartfracVs?  r&zlist[int] | None)nrcCsFg}dD]8}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  r0r)rcCstjS)z-Computes $\cos \frac{\pi}{3}$ in square roots)rHalfrrrrcos_3sr2cCstdddS)z-Computes $\cos \frac{\pi}{5}$ in square rootsr)rr rrrrcos_5sr4c Cs|tdtddtdtdtdttddtdtddtdtdtddtdddS) z.Computes $\cos \frac{\pi}{17}$ in square rootsr* rir"r rrrrcos_17s"$ r9c)Csdddddd}dddddd}|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 |'}(ttd t|(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 rztuple[Expr, Expr])r$brcSs0|t|d|d|t|d|dfSNrr r$r:rrrf1szcos_257..f1cSs|t|d|dSr;r r<rrrf2szcos_257..f2@r3r)r)rZ NegativeOnerr r1))r=r>t1t2Zz1Zz3Zz2Zz4y1Zy5Zy6y2Zy3Zy7Zy8Zy4x1Zx9Zx2x10Zx3Zx11Zx4Zx12Zx5Zx13Zx6Zx14Zx15Zx7Zx8Zx16Zv1Zv2Zv3Zv4Zv5Zv6u1u2Zw1rrrcos_257s6 """"""""rLzdict[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+)r2r4r9rLrrrr cos_tables rMN)__doc__ __future__rtypingr functoolsrZsympy.core.exprrZsympy.core.singletonrZsympy.core.intfuncrZsympy.core.numbersrZ(sympy.functions.elementary.miscellaneousr Zsympy.core.cacher rr&r0r2r4r9rLrMrrrrs*"         (K*