Similarity in hypergeometric sense means that a quotient ofį(k) and g(k) is a rational function in k. Returns True if f and g are hyper-similar. Journal of Symbolic Computation (1995) 20, 399-417 Koepf, Algorithms for m-fold Hypergeometric Summation, Otherwise Noneįor more information on the implemented algorithm refer to: Quotient of polynomials of minimal degree. If f(k) is hypergeometric then as result we arrive with a Perform simplification of nested fractions, powersĪnd if the resulting expression is a quotient of Of gamma and rising factorial with integer, absolute Rewrite all occurrences of gamma in terms of products Rewrite all functions in terms of gamma, if possible. The algorithm performs three basic steps: Integer sequences which have equivalent representation in terms The input term can be composed of functions and Given combinatorial term f(k) simplify its consecutive term ratio > from sympy import besselj, besseli, besselsimp, polar_lift, I, S > from sympy.abc import z, nu > besselsimp ( besselj ( nu, z * polar_lift ( - 1 ))) exp(I*pi*nu)*besselj(nu, z) > besselsimp ( besseli ( nu, z * polar_lift ( - I ))) exp(-I*pi*nu/2)*besselj(nu, z) > besselsimp ( besseli ( S ( - 1 ) / 2, z )) sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z)) > besselsimp ( z * besseli ( 0, z ) + z * ( besseli ( 2, z )) / 2 + besseli ( 1, z )) 3*z*besseli(0, z)/2. Separated expressions may not necessarily be grouped together.Īlthough factoring is necessary to separate variables in someĮxpressions, it is not necessary in all cases, so one should notĬount on the returned factors being factored. The order of the factors is determined by Mul, so that the If force=True, then bases of powers will be separated regardless (Passing None for symbols will return theĮxpression in a dictionary keyed to ‘coeff’.) Other symbols or non-symbols will be returned keyed to the Keys if symbols are provided, then all those symbols willīe used as keys, and any terms in the expression containing In a dictionary keyed to their corresponding symbols.īy default, all symbols in the expression will appear as If dict=True then the separated terms will be returned Byĭefault, it separates with respect to all symbols in anĮxpression and collects constant coefficients that are Separates variables in an expression, if possible. separatevars ( expr, symbols =, dict = False, force = False ) # Query for boolean expressions using assumptions. This inconsistency will be resolved in future version. Not reduce it and you should use refine() or ask()įunction. If the expression is notĮvaluated by default (such as Predicate()), simplification will If the expression prefers automatic evaluation (such asĮq() or Or()), simplification will return True orįalse if truth value can be determined. You can avoid this behavior by passing doit=False asĪlso, it should be noted that simplifying a boolean expression is not Note that simplify() automatically calls doit() on the finalĮxpression. X belongs to the set where this relation is true. If inverse=True, it will be assumed that a composition of inverseįunctions, such as sin and asin, can be cancelled in any order.įor example, asin(sin(x)) will yield x without checking whether If rational=False(default) then nothing will be done If rational=None, Floats will be recast as Rationals but the result willīe recast as Floats. If rational=True, Floats will be recast as Rationals before simplification. Simplification strategies and then compares them using the measureįunction, we get a completely different result that is still different Note that because simplify() internally tries many different 8 > 1.7 # 1.7 is the default ratio True > simplify ( g, measure = my_measure ) -log(a)*log(b) + log(a) + log(b) return count > my_measure ( g ) 8 > my_measure ( h ) 14 > 15. # Every other operation gets a weight of 1 (the default). count = count_ops ( expr, visual = True ). # Discourage powers by giving POW a weight of 10. The irrational numbers together with the rational numbers constitutes the real numbers.> from sympy import Symbol, S > def my_measure ( expr ). The decimal form of an irrational number will neither terminate nor repeat. This means that they can't be written as the quotient of two integers. The square roots of whole numbers that are not a perfect square are members of the irrational numbers. In the first section of Algebra 1 we learned that
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